Solution: The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element $a_{j i}$ shifted to new position $a_{j i}$. The symmetric matrix is defined...
If $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$ and $f(x)=x^{2}-4 x+1$, find $f(A)$.
Solution: We have $A=\left(\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right)$. Now addition/subtraction of two matrices is possible if order of both the matrices are same and multiplication...
If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.
Solution: We have $A$ and $B$ are symmetric matrices. Therefore $A^{T}=A$ and $B^{T}=B$ The transpose of the matrix is an operation of making interchange of elements by the rule on positioned...
If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$, show that $A^{\prime} A=I$.
Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$. The transpose of the matrix is an operation of making interchange of...
If $\mathrm{A}=\left[\begin{array}{ll}4 & 2 \\ 1 & 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}-2 & 1 \\ 3 & 2\end{array}\right]$, find a matrix $\mathrm{x}$ such that $3 A-2 B+X=0 .$
Solution: We have $A=\left(\begin{array}{ll}4 & 2 \\ 1 & 3\end{array}\right), B=\left(\begin{array}{cc}-2 & 1 \\ 3 & 2\end{array}\right)$ and $3 A-2 B+X=0$ We can have the addition...
If $A=\left[\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right]$ and $B=\left[\begin{array}{cc}-1 & 2 \\ 0 & 3\end{array}\right]$, find a matrix $x$ such that $A+2 B+x=0$.
Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right), B=\left(\begin{array}{cc}-1 & 2 \\ 0 & 3\end{array}\right)$ and $A+2 B+X=0$. We can have the addition...
If $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right]$, and show that $\left(\mathrm{A}-\mathrm{A}^{\prime}\right)$ is skew-symmetric
Solution: We have $\left(\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...
If $\mathrm{A}=\left[\begin{array}{ll}4 & 5 \\ 1 & 8\end{array}\right]$, show that $\left(\mathrm{A}+\mathrm{A}^{\prime}\right)$ is symmetric
Solution: We have $\left(\begin{array}{ll}4 & 5 \\ 1 & 8\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...
Find the value of $x$ and $y$ for which $\left[\begin{array}{cc} \mathrm{x} & \mathrm{y} \\ 3 \mathrm{y} & \mathrm{x} \end{array}\right]\left[\begin{array}{l} 1 \\ 2 \end{array}\right]=\left[\begin{array}{l} 3 \\ 5 \end{array}\right]$
Solution: We have $\left(\begin{array}{cc}x & y \\ 3 y & x\end{array}\right)\left(\begin{array}{l}1 \\ 2\end{array}\right)=\left(\begin{array}{l}3 \\ 5\end{array}\right)$. Use the...
Find the value of $x$ and $y$ for which $\left[\begin{array}{cc} 2 & -3 \\ 1 & 1 \end{array}\right]\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]=\left[\begin{array}{l} 1 \\ 3 \end{array}\right]$
Solution: We have $\left(\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}1 \\ 3\end{array}\right)$. Use the...
If $A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$ then find the least value of $\alpha$ for which $A+A^{\prime}=1$.
Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right)$ Use the addition rule and get $A+A^{T}=I_{2}$ as follow:...
If $\mathrm{A}=\left[\begin{array}{cr}1 & -5 \\ -3 & 2 \\ 4 & -2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}3 & 1 \\ 2 & -1 \\ -2 & 3\end{array}\right]$, find the matrix $\mathrm{C}$ such that $\mathrm{A}+\mathrm{B}+\mathrm{C}$ is a zero matrix
Solution: We have $A=\left(\begin{array}{cc}1 & -5 \\ -3 & 2 \\ 4 & -2\end{array}\right) ; B=\left(\begin{array}{cc}3 & 1 \\ 2 & -1 \\ -2 & 3\end{array}\right) .$ and...
Show that
$\begin{array}{l} \cos \theta \cdot\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \\ {\left[\begin{array}{cc} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]=I} \end{array}$
Solution: We have $\cos \theta\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right)+\sin \theta\left(\begin{array}{cc}\sin \theta & -\cos...
If A = diag (3 -2, 5) and B = diag (1 3 -4), find (A + B).
Solution: We have $A=\operatorname{diag}(3-25)=\left(\begin{array}{ccc}3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 5\end{array}\right) ; B=$...
If $\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]=3\left[\begin{array}{cc}x & y \\ z & w\end{array}\right]$, find the values of $x, y, z, \omega .$
Solution: We have $\left(\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right)+\left(\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right)=3\left(\begin{array}{cc}x & y \\ z &...
If $\left[\begin{array}{cc}x & 3 x-y \\ 2 x+z & 3 y-w\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right]$, find the values of $x, y, z, \omega$.
Solution: We have $\left(\begin{array}{cc}x & 3 x-y \\ 2 x+z & 3 y-w\end{array}\right)=\left(\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right)$. Now from the equality of matrices we...
If $x \cdot\left[\begin{array}{l}2 \\ 3\end{array}\right]+y \cdot\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{r}10 \\ 5\end{array}\right]$, find the values of $x$ and $y$.
Solution: We have $x\left(\begin{array}{l}2 \\ 3\end{array}\right)+y\left(\begin{array}{c}-1 \\ 1\end{array}\right)=\left(\begin{array}{c}10 \\ 5\end{array}\right)$. Use the addition rule and get...
Find the values of $x$ and $y$, if $2\left[\begin{array}{ll} 1 & 3 \\ 0 & x \end{array}\right]+\left[\begin{array}{ll} y & 0 \\ 1 & 2 \end{array}\right]=\left[\begin{array}{ll} 5 & 6 \\ 1 & 8 \end{array}\right]$
Solution: We have $2\left(\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right)+\left(\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right)=\left(\begin{array}{ll}5 & 6 \\ 1 &...
If $\left[\begin{array}{cc}x+2 y & -y \\ 3 x & 4\end{array}\right]=\left[\begin{array}{cc}-4 & 3 \\ 6 & 4\end{array}\right]$, find the values of $x$ and $y$
Solution: We have $\left(\begin{array}{cc}x+2 y & -y \\ 3 x & 4\end{array}\right)=\left(\begin{array}{cc}-4 & 3 \\ 6 & 4\end{array}\right)$ Now from the equality of matrices we can...
Construct a $2 \times 3$ matrix whose elements are given by $a_{i j}=\frac{1}{2}|-3 i+j|$
Solution: We have $a_{i j}=\frac{1}{2}|-3 i+j|^{2}$ Now $\begin{array}{l} a_{11}=\frac{|-3(1)+1|}{2}=1, a_{12}=\frac{|-3(1)+2|}{2}=\frac{9}{2}, a_{13}=\frac{|-3(1)+3|}{2}=\frac{9}{2} \\...
Construct a $3 \times 2$ matrix whose elements are given by $a_{i j}=\frac{1}{2}(i-2 j)^{2}$
Solution: We have $a_{i j}=\frac{1}{2}(i-2 j)^{2}$ Now $\begin{array}{l} a_{11}=\frac{(1-2(1))^{2}}{2}=\frac{1}{2}, a_{12}=\frac{(1-2(2))^{2}}{2}=\frac{9}{2} \\ a_{21}=\frac{(2-2(1))^{2}}{2}=0...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} -1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ccc}-1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} 3 & 0 & -1 \\ 2 & 3 & 0 \\ 0 & 4 & 1 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ccc}1 & 2 & 3 \\ 2 & 5 & 7 \\ -2 & -4 & -5\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 5 & 7 \\ -2 & -4 & -5 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ccc}1 & 2 & 3 \\ 2 & 5 & 7 \\ -2 & -4 & -5\end{array}\right)$. To get the inverse we will proceed by augmented matrix with elementary...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ccc}1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} 3 & -1 & -2 \\ 2 & 0 & -1 \\ 3 & -5 & 0 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & -1 \\ 3 & -5 & 0\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ccc}1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{lll} 3 & 0 & 2 \\ 1 & 5 & 9 \\ 6 & 4 & 7 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{lll}3 & 0 & 2 \\ 1 & 5 & 9 \\ 6 & 4 & 7\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ccc}2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ccc} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ll} 6 & 7 \\ 8 & 9 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ll}6 & 7 \\ 8 & 9\end{array}\right)$. To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ll} 4 & 0 \\ 2 & 5 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ll}4 & 0 \\ 2 & 5\end{array}\right)$. To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{cc} 2 & -3 \\ 4 & 5 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{rr} 2 & 5 \\ -3 & 1 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}2 & 5 \\ -3 & 1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{cc} 1 & 2 \\ 2 & -1 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
Using elementary row transformations, find the inverse of each of the following matrices: $\left[\begin{array}{ll} 1 & 2 \\ 3 & 7 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 7\end{array}\right)$ To get the inverse we will proceed by augmented matrix with elementary row transformation process is as follow:...
If matrix $A=[123]$, write $\mathrm{AA}^{\prime}$.
Solution: Given that $A=\left[\begin{array}{ll}1 & 23\end{array}\right]$ We will find $A$ ' to calculate AA': $\mathrm{A}^{\prime}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]$ Now...
If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$, show that $A^{\prime} A=1$.
Solution: Given that $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \theta & \cos \alpha\end{array}\right]$. We wil find $A$ $A^{\prime}=\left[\begin{array}{cc}\cos \alpha...
For each of the following pairs of matrices $A$ and $B$, verify that $(A B)^{\prime}=\left(B^{\prime} A^{\prime}\right)$ : $A=\left[\begin{array}{rrr}-1 & 2 & -3 \\ 4 & -5 & 6\end{array}\right]$ and $B=\left[\begin{array}{cc}3 & -4 \\ 2 & 1 \\ -1 & 0\end{array}\right]$
Solution: Take $\mathrm{C}=\mathrm{AB}$ $\begin{array}{l} C=\left[\begin{array}{ccc} -1 & 2 & -3 \\ 4 & -5 & 6 \end{array}\right]\left[\begin{array}{cc} 3 & -4 \\ 2 & 1 \\ -1...
For each of the following pairs of matrices $A$ and $B$, verify that $(A B)^{\prime}=\left(B^{-} A^{\prime}\right)$ : $\mathrm{A}=\left[\begin{array}{c} -1 \\ 2 \\ 3 \end{array}\right] \text { and } \mathrm{B}=[-2 \cdot 1-4]$
Solution: Take $C=A B$ $\begin{array}{l} C=\left[\begin{array}{c} -1 \\ 2 \\ 3 \end{array}\right]\left[\begin{array}{lll} -2 & -1 & -4 \end{array}\right] \\...
For each of the following pairs of matrices $A$ and $B$, verify that $(A B)^{\prime}=\left(B^{\prime} A^{\prime}\right)$ : $A=\left[\begin{array}{rr} 3 & -1 \\ 2 & -2 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 1 & -3 \\ 2 & -1 \end{array}\right]$
Solution: Take $C=A B$ $\begin{array}{l} C=\left[\begin{array}{rr} 3 & -1 \\ 2 & -2 \end{array}\right]\left[\begin{array}{ll} 1 & -3 \\ 2 & -1 \end{array}\right] \\...
For each of the following pairs of matrices $A$ and $B$, verify that $(A B)^{\prime}=\left(B^{\prime} A^{\prime}\right)$ : $A=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 1 & 4 \\ 2 & 5 \end{array}\right]$
Solution: Take $\mathrm{C}=\mathrm{A} 8$ $\begin{array}{l} C=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]\left[\begin{array}{ll} 1 & 4 \\ 2 & 5 \end{array}\right] \\...
Express the matrix $A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$ as sum af two matrices such that and is symmetric and the other is skew-symmetric.
Solution: Given that $A=\left[\begin{array}{lll}3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix...
Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$.
Solution: Given that $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix...
Express the matrix $\mathrm{A}=\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.
Solution: Given that $\mathrm{A}=\left[\begin{array}{ccc}-1 & 5 & 1 \\ 2 & 3 & 4 \\ 7 & 0 & 9\end{array}\right]$, to express as sum of symmetric matrix $\mathrm{P}$ and skew...
Express the matrix $\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]$ as the sum of a symmetric matrix and a skew-symmetric matrix.
Solution: Given that $\mathrm{A}=\left[\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right]$,to express as the sum of symmetric matrix $\mathrm{P}$ and skew symmetric matrix Q. $A=P+Q$ Where...
Express the matrix $\mathrm{A}=\left[\begin{array}{cc}2 & 3 \\ -1 & 4\end{array}\right]$ as the sum of a symmetric matrix and a skew-symmetric matrix.
Solution: Given that $A=\left[\begin{array}{cc}2 & 3 \\ -1 & 4\end{array}\right]$, As for a symmetric matrix $A^{\prime}=A$ hence $\begin{array}{l} A+A^{\prime}=2 A \\...
Show that the matrix $\mathrm{A}=\left[\begin{array}{ccc}0 & \mathrm{a} & \mathrm{b} \\ -\mathrm{a} & 0 & \mathrm{c} \\ -\mathrm{b} & -\mathrm{c} & 0\end{array}\right]$ is skew-symmetric.
Solution: We have $A=\left(\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right)$. The transpose of the matrix is an operation of making interchange of...
If $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$, show that $\left(A+A^{\prime}\right)$ is skew-symmetric.
Solution: We have $A=\left(\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right)$ The transpose of the matrix is an operation of making interchange of elements by the rule on positioned...
If $A=\left[\begin{array}{ll}4 & 1 \\ 5 & 8\end{array}\right]$, show that $\left(A+A^{\prime}\right)$ is symmetric.
Solution: We have $A=\left(\begin{array}{ll}4 & 1 \\ 5 & 8\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on positioned element...
If $P=\left[\begin{array}{rr}3 & 4 \\ 2 & -1 \\ 0 & 5\end{array}\right]$ and $P=\left[\begin{array}{cc}7 & -5 \\ -4 & 0 \\ 2 & 6\end{array}\right]$, verify that $(P+Q)^{\prime}=\left(P^{\prime}+Q^{\prime}\right)$
Solution: We have $P=\left(\begin{array}{cc}3 & 4 \\ 2 & -1 \\ 0 & 5\end{array}\right)$ and $Q=\left(\begin{array}{cc}7 & -5 \\ -4 & 0 \\ 2 & 6\end{array}\right)$. The...
If $A=\left[\begin{array}{rrr}3 & 2 & -1 \\ -5 & 0 & -6\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4 & -5 & -2 \\ 3 & 1 & 8\end{array}\right]$, verify that $(A+B)^{\prime}=\left(A^{\prime}+B^{\prime}\right)$.
Solution: We have $A=\left(\begin{array}{ccc}3 & 2 & -1 \\ -5 & 0 & -6\end{array}\right)$ and $B=\left(\begin{array}{ccc}-4 & -5 & -2 \\ 3 & 1 & 8\end{array}\right)$....
If $A=\left[\begin{array}{cc}3 & 5 \\ -2 & 0 \\ 4 & -6\end{array}\right]$, verify that $(2 A)^{\prime}=2 A^{\prime}$.
Solution: We have $A=\left(\begin{array}{cc}3 & 5 \\ -2 & 0 \\ 4 & -6\end{array}\right)$. Thus $2 A=\left(\begin{array}{cc}6 & 10 \\ -4 & 0 \\ 8 & -12\end{array}\right)$ The...
If $A=\left[\begin{array}{ccc}2 & -3 & 5 \\ 0 & 7 & -4\end{array}\right]$, verify that $\left(A^{\prime}\right)^{\prime}=A$.
Solution: We have $A=\left(\begin{array}{ccc}2 & -3 & 5 \\ 0 & 7 & -4\end{array}\right)$. The transpose of the matrix is an operation of making interchange of elements by the rule on...
If $\left[\begin{array}{cc}2 & 3 \\ 5 & 7\end{array}\right]\left[\begin{array}{cc}1 & -3 \\ -2 & 4\end{array}\right]=\left[\begin{array}{cc}-4 & 6 \\ -9 & \mathrm{x}\end{array}\right]$, find the value of $\mathrm{x}$
Solution: We have $A=\left(\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right), B=\left(\begin{array}{cc}1 & -3 \\ -2 & 4\end{array}\right)$ and $C=\left(\begin{array}{cc}-4 & 6 \\...
If $A=\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]$, find $\left(3 A^{2}-2 B+1\right)$.
Solution: We have $A=\left(\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right), B=\left(\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right)$. (i) Let's compute first $A^{2}$ $A^{2}=A...
Give an example of three matrices A, B, C such that AB = AC but B ≠ C.
Solution: We have $\boldsymbol{A B}=\boldsymbol{A} \boldsymbol{C}$ but $\boldsymbol{B} \neq \boldsymbol{C}$. WE need to find: $\boldsymbol{A}, \boldsymbol{B}$. Let's take...
Given an example of two matrices A and B such that A ≠ O, B ≠ O, AB = O and BA ≠ O.
Solution: We have $\boldsymbol{A} \neq \boldsymbol{O}, \boldsymbol{B} \neq \boldsymbol{O}, \boldsymbol{A B}=\boldsymbol{O}$ and $\boldsymbol{B A} \neq \boldsymbol{O}$. We need to find:...
If $\mathrm{A}=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$, prove that $\mathrm{A}^{\mathrm{n}}=\left[\begin{array}{ll}1 & \mathrm{n} \\ 0 & 1\end{array}\right]$ for all $\mathrm{n} \in \mathrm{N}$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{1} & \mathbf{1} \\ \mathbf{0} & \mathbf{1}\end{array}\right)$. We need to show: $A^{n}=\left(\begin{array}{ll}1 & n \\ 0...
If $A=\left[\begin{array}{cc}3 & 4 \\ -4 & -3\end{array}\right]$, find $f(A)$, where $f(x)=x 2-5 x+7$.
Solution: We have $A=\left(\begin{array}{cc}3 & 4 \\ -4 & -3\end{array}\right)$ and equation $f(x)=x^{2}-5 x+7$. (i) Let us compute first $A^{2}$ $A^{2}=A A=\left(\begin{array}{cc} 3 & 4...
Find the values of $a$ and $b$ for which $\left[\begin{array}{cc} a & b \\ -a & 2 b \end{array}\right]\left[\begin{array}{c} 2 \\ -1 \end{array}\right]=\left[\begin{array}{l} 5 \\ 4 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\boldsymbol{a} & \boldsymbol{b} \\ -\boldsymbol{a} & \boldsymbol{2} \boldsymbol{b}\end{array}\right),...
If $\left[\begin{array}{lll}x & 4 & 1\end{array}\right]\left[\begin{array}{ccc}2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & -4\end{array}\right]\left[\begin{array}{c}x \\ 4 \\ -1\end{array}\right]=\mathrm{O}$, find $x$
Solution: We have $A=\left(\begin{array}{ccc}2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & -4\end{array}\right), B=\left(\begin{array}{lll}x & 4 & 1\end{array}\right)$ and...
If $\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5\end{array}\right]\left[\begin{array}{c}1 \\ -2 \\ 3\end{array}\right]=\mathrm{O}$, find $x$
Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5\end{array}\right), B=\left(\begin{array}{lll}1 & x & 1\end{array}\right)$ and...
If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$, show that $A^{2}=\left[\begin{array}{cc}\cos 2 \alpha & \sin 2 \alpha \\ -\sin 2 \alpha & \cos 2 \alpha\end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$ and to show $A^{2}=$ $\left(\begin{array}{cc}\cos ^2 \alpha & \sin...
If $F(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$, show that $F(x) \cdot F(y)=F(x+y)$
Solution: We have $\boldsymbol{F}(\boldsymbol{X})=\left(\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right)$ and to show...
If $A=\left[\begin{array}{rr}1 & -1 \\ 2 & -1\end{array}\right], B=\left[\begin{array}{rr}a & -1 \\ b & -1\end{array}\right]$ and $(A+B)^{2}=\left(A^{2}+B^{2}\right)$ then find the values of $a$ and $b$.
Solution: We have $A=\left(\begin{array}{ll}1 & -1 \\ 2 & -1\end{array}\right), B=\left(\begin{array}{ll}a & -1 \\ b & -1\end{array}\right)$ and $(A+B)^{2}=\left(A^{2}+\right.$...
Find the matrix A such that A. $\left[\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right]=\left[\begin{array}{cc}0 & -4 \\ 10 & 3\end{array}\right]$.
Solution: We have $\boldsymbol{B}=\left(\begin{array}{ll}\mathbf{2} & \mathbf{3} \\ \mathbf{4} & \mathbf{5}\end{array}\right)$ and $\boldsymbol{C}=\left(\begin{array}{cc}\mathbf{0} &...
Find the matrix A such that $\left[\begin{array}{cc}5 & -7 \\ -2 & 3\end{array}\right] \cdot \mathrm{A}=\left[\begin{array}{cc}-16 & -6 \\ 7 & 2\end{array}\right]$.
Solution: We have $\boldsymbol{B}=\left(\begin{array}{cc}\mathbf{5} & -\mathbf{7} \\ -\mathbf{2} & \mathbf{3}\end{array}\right)$ and $\boldsymbol{C}=\left(\begin{array}{cc}-\mathbf{1 6}...
If $A=\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right]$, find the value of a and $b$ such that $A^{2}+a A+b l=0$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & \mathbf{2} \\ \mathbf{1} & \mathbf{1}\end{array}\right)$. To find $\boldsymbol{a}, \boldsymbol{b}$ such that...
If $A=\left[\begin{array}{ll}3 & 1 \\ 7 & 5\end{array}\right]$, find $x$ and $y$ such that $A^{2}+x \mid=y A$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & \mathbf{1} \\ \mathbf{7} & \mathbf{5}\end{array}\right)$. To find $\boldsymbol{x}, \boldsymbol{y}$ such that...
Solve for $x$ and $y$, when $\left[\begin{array}{cc} 3 & -4 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]=\left[\begin{array}{l} 3 \\ 11 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}3 & -4 \\ 1 & 2\end{array}\right), B=\left(\begin{array}{c}3 \\ 11\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y\end{array}\right)$. We...
Find the values of $x$ and $y$, when $\left[\begin{array}{cc} 2 & -3 \\ 1 & 1 \end{array}\right]\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \end{array}\right]=\left[\begin{array}{l} 1 \\ 3 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right), B=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y\end{array}\right)$. To...
If $A=\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]$ and $f(x)=2 x^{3}+4 x+5$, find $f(A)$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{1} & \mathbf{2} \\ \mathbf{4} & \mathbf{- 3}\end{array}\right)$. Now addition/subtraction of two matrices is possible if...
If $A=\left[\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right]$, find $f(A)$, where $f(x)=x^{2}-2 x+3$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)$. Now addition/subtraction of two matrices is possible if order of both the matrices are same and...
If $A=\left[\begin{array}{rr}3 & -2 \\ 4 & -2\end{array}\right]$, find $k$ so that $A^{2}=k A-21$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & -\mathbf{2} \\ \mathbf{4} & -\mathbf{2}\end{array}\right)$. Now addition/subtraction of two matrices is possible if...
Show that the matrix $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$ satisfies the equation $\mathrm{A}^{3}-4 \mathrm{~A}^{2}+\mathrm{A}=0$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{2} & \mathbf{3} \\ \mathbf{1} & \mathbf{2}\end{array}\right)$. Now addition of two matrices is possible if order of both the...
If $A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]$, show that $(A 2-5 A+71)=0$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{3} & \mathbf{1} \\ -\mathbf{1} & \mathbf{2}\end{array}\right)$. Now addition of two matrices is possible if order of both the...
If $A=\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$ then find $\left(-A^{2}+6 A\right)$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 2} \\ -\mathbf{3} & \mathbf{4}\end{array}\right)$. Now addition of two matrices is possible if order of both...
If $A=\left[\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right]$, find $\left(3 A^{2}-2 B+1\right)$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{2}\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{cc}\mathbf{0} &...
If $A=\left[\begin{array}{ccc}4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3\end{array}\right]$, show that $A^{2}=1$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{4} & -\mathbf{1} & -\mathbf{4} \\ \mathbf{3} & \mathbf{0} & -\mathbf{4} \\ \mathbf{3} & -\mathbf{1} &...
If $A=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]$, show that $A^{2}=A$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & \mathbf{- 2} & -\mathbf{4} \\ \mathbf{- 1} & \mathbf{3} & \mathbf{4} \\ \mathbf{1} & -\mathbf{2} &...
If $A=\left[\begin{array}{cc}a b & b^{2} \\ -a^{2} & -a b\end{array}\right]$, show that $A^{2}=0$.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\boldsymbol{a} \boldsymbol{b} & \boldsymbol{b}^{2} \\ -\boldsymbol{a}^{2} & -\boldsymbol{a} \boldsymbol{b}\end{array}\right) .$ To...
If $A=\left[\begin{array}{ccc}1 & 0 & -2 \\ 3 & -1 & 0 \\ -2 & 1 & 1\end{array}\right], B=\left[\begin{array}{ccc}0 & 5 & -4 \\ -2 & 1 & 3 \\ 1 & 0 & 2\end{array}\right]$ and $C=\left[\begin{array}{ccc}1 & 5 & 2 \\ -1 & 1 & 0 \\ 0 & -1 & 1\end{array}\right] ;$ verify that $A(B-C)=(A B-A C)$
Solution: We have $A=\left(\begin{array}{ccc}1 & 0 & 2 \\ 3 & -1 & 0 \\ -2 & 1 & 1\end{array}\right), B=\left(\begin{array}{ccc}0 & 5 & -4 \\ -2 & 1 & 3 \\ 1...
Verify that $A(B+C)=(A B+A C)$, when $\mathrm{A}=\left[\begin{array}{cc} 2 & 3 \\ -1 & 4 \\ 0 & 1 \end{array}\right], \mathrm{B}=\left[\begin{array}{cc} 5 & -3 \\ 2 & 1 \end{array}\right] \text { and } \mathrm{C}=\left[\begin{array}{cc} -1 & 2 \\ 3 & 4 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}2 & 3 \\ -1 & 4 \\ 0 & 1\end{array}\right), B=\left(\begin{array}{cc}5 & -3 \\ 2 & 1\end{array}\right)$. and...
Verify that $A(B+C)=(A B+A C)$, when $\mathrm{A}=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \mathrm{B}=\left[\begin{array}{cc} 2 & 0 \\ 1 & -3 \end{array}\right] \text { and } \mathrm{C}=\left[\begin{array}{cc} 1 & -1 \\ 0 & 1 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{1} & \mathbf{2} \\ \mathbf{3} & \mathbf{4}\end{array}\right), \boldsymbol{B}=\left(\begin{array}{cc}\mathbf{2} &...
For the following matrices, verify that $A(B C)=(A B) C$ : $\mathrm{A}=\left[\begin{array}{ccc} 2 & 3 & -1 \\ 3 & 0 & 2 \end{array}\right], \mathrm{B}=\left[\begin{array}{l} 1 \\ 1 \\ 2 \end{array}\right] \text { and } \mathrm{C}=\left[\begin{array}{ll} 1 & -2 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & \mathbf{3} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{0} & \mathbf{2}\end{array}\right),...
For the following matrices, verify that $A(B C)=(A B) C$ : $\mathrm{A}=\left[\begin{array}{lll} 1 & 2 & 5 \\ 0 & 1 & 3 \end{array}\right], \mathrm{B}=\left[\begin{array}{lll} 2 & 3 & 0 \\ 1 & 0 & 4 \\ 1 & -1 & 2 \end{array}\right] \text { and } \mathrm{C}=\left[\begin{array}{l} 1 \\ 4 \\ 5 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 5 \\ 0 & 1 & 3\end{array}\right), B=\left(\begin{array}{ccc}2 & 3 & 0 \\ 1 & 0 & 4 \\ 1 & -1 &...
If $A=\left[\begin{array}{ccc}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right]$ and $B=\left[\begin{array}{lll}a^{2} & a b & a c \\ a b & b^{2} & b c \\ a c & b c & c^{2}\end{array}\right]$. show that $A B$ is a zero matrix.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \boldsymbol{c} & -\boldsymbol{b} \\ -\boldsymbol{c} & \mathbf{0} & \boldsymbol{a} \\ \boldsymbol{b} &...
$\text { If } A=\left[\begin{array}{ccc} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{array}\right] \text { and } B=\left[\begin{array}{ccc} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{array}\right] \text {, shown that } A B=A \text { and } B A=B \text {. }$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & -\mathbf{3} & -\mathbf{5} \\ -\mathbf{1} & \mathbf{4} & \mathbf{5} \\ \mathbf{1} & -\mathbf{3} &...
Show that $A B=B A$ in each of the following cases: $A=\left[\begin{array}{ccc} 1 & 3 & -1 \\ 2 & 2 & -1 \\ 3 & 0 & -1 \end{array}\right] \text { and } B=\left[\begin{array}{ccc} -2 & 3 & -1 \\ -1 & 2 & -1 \\ -6 & 9 & -4 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{1} & \mathbf{3} & -\mathbf{1} \\ \mathbf{2} & \mathbf{2} & -\mathbf{1} \\ \mathbf{3} & \mathbf{0} &...
Show that $A B=B A$ in each of the following cases: $\mathrm{A}=\left[\begin{array}{lll} 1 & 2 & 1 \\ 3 & 4 & 2 \\ 1 & 3 & 2 \end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{ccc} 10 & -4 & -1 \\ -11 & 5 & 0 \\ 9 & -5 & 1 \end{array}\right]$
Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 1 \\ 3 & 4 & 2 \\ 1 & 3 & 2\end{array}\right)$ and $B=\left(\begin{array}{ccc}10 & -4 & -1 \\ -11 & 5 & 0...
Show that $A B=B A$ in each of the following cases: $A=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] \text { and } B=\left[\begin{array}{cc} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right]$
Solution: We have $A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$ and $B=\left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi...
Show that $A B \neq B A$ in each of the following cases: $\mathrm{A}=\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{ccc} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{lll}\mathbf{1} & \mathbf{2} & \mathbf{3} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{1} & \mathbf{1} &...
Show that $A B \neq B A$ in each of the following cases : $A=\left[\begin{array}{cc} 5 & -1 \\ 6 & 7 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 2 & 1 \\ 3 & 4 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{5} & -\mathbf{1} \\ \mathbf{6} & \mathbf{7}\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{ll}\mathbf{2} &...
Compute $A B$ and $B A$, which ever exists when $A=\left[\begin{array}{cc} 2 & 1 \\ 3 & 2 \\ -1 & 1 \end{array}\right] \text { and } B=\left[\begin{array}{rrr} 1 & 0 & 1 \\ -1 & 2 & 1 \end{array}\right]$
Solution: We have $\boldsymbol{B}=\left(\begin{array}{ccc}\mathbf{1} & \mathbf{0} & \mathbf{1} \\ -\mathbf{1} & \mathbf{2} & \mathbf{1}\end{array}\right)$ and...
Compute $A B$ and BA, which ever exists when $A=\left[\begin{array}{lll} 1 & 2 & 3 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{llll}\mathbf{1} & 2 & 3 & 4\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{l}\mathbf{1} \\ \mathbf{2} \\ \mathbf{3} \\...
Compute AB and BA, which ever exists when $\mathrm{A}=\left[\begin{array}{rrr} 0 & 1 & -5 \\ 2 & 4 & 0 \end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{cc} 1 & 3 \\ -1 & 0 \\ 0 & 5 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \mathbf{1} & -\mathbf{5} \\ \mathbf{2} & \mathbf{4} & \mathbf{0}\end{array}\right)$ and...
Compute AB and BA, which ever exists when $\mathrm{A}=\left[\begin{array}{cc} -1 & 1 \\ -2 & 2 \\ -3 & 3 \end{array}\right] \text { and } \mathrm{B}=\left[\begin{array}{ccc} 3 & -2 & 1 \\ 0 & 1 & 2 \\ -3 & 4 & -5 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}-\mathbf{1} & \mathbf{1} \\ -\mathbf{2} & \mathbf{2} \\ -\mathbf{3} & \mathbf{3}\end{array}\right)$ and...
Compute AB and BA, which ever exists when $A=\left[\begin{array}{cc} 2 & -1 \\ 3 & 0 \\ -1 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{cc} -2 & 3 \\ 0 & 4 \end{array}\right]$
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{0} \\ \mathbf{- 1} & \mathbf{4}\end{array}\right)$ and...
If $\left[\begin{array}{cc}x-y & 2 y \\ 2 y+z & x+y\end{array}\right]=\left[\begin{array}{ll}1 & 4 \\ 9 & 5\end{array}\right]$ then write the value of $(x+y)$
Solution: $\text { If }\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} e & f \\ g & h \end{array}\right]$ Therefore $a=e, b=f, c=g, d=h$ It is given...
Find the value of $(x+y)$ from the following equation : $2\left[\begin{array}{ll} 1 & 3 \\ 0 & \mathrm{x} \end{array}\right]+\left[\begin{array}{ll} \mathrm{y} & 0 \\ 1 & 2 \end{array}\right]=\left[\begin{array}{ll} 5 & 6 \\ 1 & 8 \end{array}\right]$
Solution: It is given that $\begin{array}{l} 2\left[\begin{array}{ll} 1 & 3 \\ 0 & x \end{array}\right]+\left[\begin{array}{ll} y & 0 \\ 1 & 2...
Find the value of $x$ and $y$, when
i. $2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{cc}3 & -4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 6 \\ 15 & 14\end{array}\right]$
Solution: (i) Given $2\left(\begin{array}{lc} x & 5 \\ 7 y-3 \end{array}\right)+\left(\begin{array}{c} 3-4 \\ 12 \end{array}\right)=\left(\begin{array}{cc} 7 & 6 \\ 1514 \end{array}\right)$...
Find the value of $x$ and $y$, when
i. $\left[\begin{array}{l}\mathrm{x}+\mathrm{y} \\ \mathrm{x}-\mathrm{y}\end{array}\right]=\left[\begin{array}{l}8 \\ 4\end{array}\right]$
ii. $\left[\begin{array}{cc}2 x+5 & 7 \\ 0 & 3 y-7\end{array}\right]=\left[\begin{array}{cc}x-3 & 7 \\ 0 & -5\end{array}\right]$
Solution: (i) Given $\left(\begin{array}{l}\boldsymbol{x}+\boldsymbol{y} \\ \boldsymbol{x}-\boldsymbol{y}\end{array}\right)=\left(\begin{array}{l}8 \\ \mathbf{4}\end{array}\right)$. By equality of...
If $A=\operatorname{diag}[2,-5,9], B=\operatorname{diag}[-3,7,14]$ and $C=\operatorname{diag}[4,-6,3]$, find:
(i) $A+2 B$
(ii) $\mathrm{B}+\mathrm{C}-\mathrm{A}$
Solution: If $Z=\operatorname{diag}[a, b, c]$, then it can be written as $Z=\left[\begin{array}{lll} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right]$ Therefore, $A+2...
If $A=\left[\begin{array}{rrr}1 & -3 & 2 \\ 2 & 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1\end{array}\right]$, find a matrix $C$ such that $(A+B+C)$ is a zero matrix.
Solution: It is given that $A+B+C$ is zero matrix i.e $A+B+C=0$ $\begin{array}{l} {\left[\begin{array}{ccc} 1 & -3 & 2 \\ 2 & 0 & 2 \end{array}\right]+\left[\begin{array}{ccc} 2...
Find the matrix $X$ such that $2 A-B+X=0$ where $A=\left[\begin{array}{ll}3 & 1 \\ 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{cc}-2 & 1 \\ 0 & 3\end{array}\right]$
Solution: It is given that $2 A-B+X=0$ $\begin{array}{l} 2\left(\left[\begin{array}{ll} 3 & 1 \\ 0 & 2 \end{array}\right]\right)-\left[\begin{array}{cc} -2 & 1 \\ 0 & 3...
If $A=\left[\begin{array}{cc}-2 & 3 \\ 4 & 5 \\ 1 & -6\end{array}\right]$ and $B=\left[\begin{array}{cc}5 & 2 \\ -7 & 3 \\ 6 & 4\end{array}\right]$, find a matrix $C$ such that $A+B-C=0$
Solution: It is given that $A+B-C=0$ $\begin{array}{c} {\left[\begin{array}{cc} -2 & 3 \\ 4 & 5 \\ 1 & -6 \end{array}\right]+\left[\begin{array}{cc} 5 & 2 \\ -7 & 3 \\ 6 & 4...
Find matrix $X$, if $\left[\begin{array}{rrr}3 & 5 & -9 \\ -1 & 4 & -7\end{array}\right]+X=\left[\begin{array}{lll}6 & 2 & 3 \\ 4 & 8 & 6\end{array}\right]$
Solution: It is given that $\left[\begin{array}{ccc}3 & 5 & -9 \\ -1 & 4 & -7\end{array}\right]+x=\left[\begin{array}{lll}6 & 2 & 3 \\ 4 & 8 & 6\end{array}\right]$...
Find matrices $A$ and $B$, if $2 A-B=\left[\begin{array}{rrr}6 & -6 & 0 \\ -4 & 2 & 1\end{array}\right]$ and $2 B+A=\left[\begin{array}{ccc}3 & 2 & 5 \\ -2 & 1 & -7\end{array}\right]$
Solution: $\operatorname{Add} 2(2 A-B)$ and $(2 B+A)$ $\begin{array}{l} 2(2 A-B)+(2 B+A)=2\left(\left[\begin{array}{ccc} 6 & -6 & 0 \\ -4 & 2 & 1...
Find matrices $A$ and $B$, if $A+B=\left[\begin{array}{ccc}1 & 0 & 2 \\ 5 & 4 & -6 \\ 7 & 3 & 8\end{array}\right]$ and $A-B=\left[\begin{array}{ccc}-5 & -4 & 8 \\ 11 & 2 & 0 \\ -1 & 7 & 4\end{array}\right]$
Solution: Add $(A+B)$ and $(A-B)$ We obtain $(A+B)+(A-B)=\left[\begin{array}{ccc}1 & 0 & 2 \\ 5 & 4 & -6 \\ 7 & 3 & 8\end{array}\right]+\left[\begin{array}{ccc}-5 & -4...
If $5 \mathrm{~A}=\left[\begin{array}{ccc}5 & 10 & -15 \\ 2 & 3 & 4 \\ 1 & 0 & -5\end{array}\right]$, find $\mathrm{A}$
Solution: $5 A=\left[\begin{array}{ccc}5 & 10 & -15 \\ 2 & 3 & 4 \\ 1 & 0 & -5\end{array}\right]$ $A=\left[\begin{array}{ccc}\frac{5}{5} & \frac{10}{5} &...
Let $A=\left[\begin{array}{ccc}0 & 1 & -2 \\ 5 & -1 & -4\end{array}\right], B=\left[\begin{array}{ccc}1 & -3 & -1 \\ 0 & -2 & 5\end{array}\right]$ and $C=\left[\begin{array}{ccc}2 & -5 & 1 \\ -4 & 0 & 6\end{array}\right] .$ Compute $5 A-3 B+4 C$
Solution: $\begin{array}{l} \left.5 A-3 B+4 C=5\left(\left[\begin{array}{ccc} 0 & 1 & -2 \\ 5 & -1 & -4 \end{array}\right]\right)-3\left(\begin{array}{ccc} 1 & -3 & -1 \\ 0...
Let $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$ and $C=\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right] .$ Find:
i. $A-2 B+3 C$
Solution: $\begin{array}{l} \text { i. } A-2 B+3 C \\ \text { A- } 2 B+3 C=\left[\begin{array}{ll} 2 & 4 \\ 3 & 2 \end{array}\right]-2\left(\left[\begin{array}{cc} 1 & 3 \\ -2 & 5...
Let $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$ and $C=\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right] .$ Find:
i. $A+2 B$
ii. B – $4 c$
Solution: i. $\begin{array}{l} A+2 B=\left[\begin{array}{ll} 2 & 4 \\ 3 & 2 \end{array}\right]+2\left(\left[\begin{array}{cc} 1 & 3 \\ -2 & 5 \end{array}\right]\right) \\...
If $A=\left[\begin{array}{ccc}3 & 1 & 2 \\ 1 & 2 & -3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2 & 0 & 4 \\ 5 & -3 & 2\end{array}\right]$, find $(2 A-B)$
Solution: $\begin{array}{l} 2 A=2\left(\left[\begin{array}{ccc} 3 & 1 & 2 \\ 1 & 2 & -3 \end{array}\right]\right) \\ =\left[\begin{array}{ccc} 6 & 2 & 4 \\ 2 & 4 & -6...
If $A=\left[\begin{array}{cc}3 & 5 \\ -2 & 0 \\ 6 & -1\end{array}\right], B=\left[\begin{array}{cc}-1 & -3 \\ 4 & 2 \\ -2 & 3\end{array}\right]$ and $C=\left[\begin{array}{cc}0 & 2 \\ 3 & -4 \\ 1 & 6\end{array}\right]$, verify that $(A+B)+C=A+(B+C)$
Solution: $\begin{array}{l} (A+B)+C=\left(\left[\begin{array}{cc} 3 & 5 \\ -2 & 0 \\ 6 & -1 \end{array}\right]+\left[\begin{array}{cc} -1 & -3 \\ 4 & 2 \\ -2 & 3...
If $A=\left[\begin{array}{ccc}2 & -3 & 5 \\ -1 & 0 & 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}3 & 2 & -2 \\ 4 & -3 & 1\end{array}\right]$, verify that $(A+B)=(B+A)$
Solution: $\begin{array}{l} A+B=\left[\begin{array}{ccc} 2 & -3 & 5 \\ -1 & 0 & 3 \end{array}\right]+\left[\begin{array}{ccc} 3 & 2 & -2 \\ 4 & -3 & 1...
Construct a $3 \times 4$ matrix whose elements are given by $a_{i j}=\frac{1}{2}|-3 i+j|$
Solution: It is a (3 $x 4)$ matrix. Therefore, it has 3 rows and 4 columns. Given that $a_{i j}=\frac{|-a \|+| l}{2}$ Therefore, $a_{11}=1 . a_{12}=\frac{1}{2}, a_{13}=0, a_{13}=\frac{1}{2}$...
Construct a $2 \times 3$ matrix whose elements are $a_{i j}=\frac{(i-2 j)^{2}}{2}$
Solution: It is a $(2 \times 3)$ matrix. Therefore, it has 2 rows and 3 columns. Given that $a_{i j}=\frac{\left(t-2 \rho^{2}\right.}{2}$ Therefore, $a_{11}=\frac{1}{2}, a_{12}=\frac{9}{2},...
Construct a $2 \times 2$ matrix whose elements are $a_{i j}=\frac{(i+2 j)^{2}}{2}$
Solution: It is a $(2 \times 2)$ matrix. So, it has 2 rows and 2 columns. Given that $a_{i j}=\frac{(i+2 j)^{2}}{2}$ Therefore, $a_{11}=\frac{9}{2}, a_{12}=\frac{25}{2}$. $a_{21}=8 . a_{22}=18$...
Construct a $4 \times 3$ matrix whose elements are given by $a_{i j}=\frac{1}{j}$
Solution: It is $(4 \times 3)$ matrix. Therefore it has 4 rows and 3 columns Given that$a_{i j}=\frac{i}{j}$ Therefore, $a_{11}=1 . a_{12}=\frac{1}{2}, a_{13}=\frac{1}{3}$ $\begin{array}{l}...
Construct a $3 \times 2$ matrix whose elements are given by $a_{j}=(2 i-j)$
Solution: It is given that: $a_{i j}=(2 \mid-j)$ Now, $a_{11}=(2 \times 1-1)=2-1=1$ $\begin{array}{l} a_{12}=2 \times 1-2=2-2=0 \\ a_{21}=2 \times 2-1=4-1=3 \\ a_{22}=2 \times 2-2=4-2=2 \\ a_{31}=2...
Find all possible orders of matrices having 7 elements.
Solution: No. of entries $=($ No. of rows) $x$ (No. of columns) $=7$ If order is $(\mathrm{a} \times \mathrm{b})$ then, No. of entries = $\mathrm{a} \times \mathrm{b}$ Therefore now a $x b=7$ (in...
If a matrix has 18 elements, what are the possible orders it can have?
Solution: No. of entries $=$ (No. of rows) $x$ (No. of columns) $=18$ If order is $(a \times b)$ then, No. of entries = $a \times b$ Therefore now $a \times b=18$ (in this case) Possible cases are...
Write the order of each of the following matrices:
i. $E=\left[\begin{array}{c}-2 \\ 3 \\ 0\end{array}\right]$
ii, $F=[6]$
Solution: i. $E=\left[\begin{array}{c}-2 \\ 3 \\ 0\end{array}\right]$ Order of matrix $=$ Number of rows $x$ Number of columns $\begin{array}{l} =(3 \times 1) \\ \text { ii, } F=[6] \end{array}$...
Write the order of each of the following matrices:
i. $\mathrm{C}=\left[\begin{array}{lll}7-\sqrt{2} & 5 & 0\end{array}\right]$
ii. $\mathrm{D}=[8-3]$
Solution: i. $C=[7-\sqrt{2} \quad 5 \quad 0]$ Order of matrix $=$ Number of rows $x$ Number of columns $=(1 \times 4)$ ii. $D=[8-3]$ Order of matrix $=$ Number of rows $x$ Number of columns $=(1...
Write the order of each of the following matrices: i. $A=\left[\begin{array}{cccc}3 & 5 & 4 & -2 \\ 0 & \sqrt{3} & -1 & \frac{4}{9}\end{array}\right]$ ii. $B=\left[\begin{array}{cc}6 & -5 \\ \frac{1}{2} & \frac{3}{4} \\ -2 & -1\end{array}\right]$
Solution: i. $A=\left[\begin{array}{cccc}3 & 5 & 4 & -2 \\ 0 & \sqrt{3} & -1 & \frac{4}{9}\end{array}\right]$ Order of matrix $=$ Number of rows $x$ Number of columns $=(2...
If $A=\left[\begin{array}{cccc}5 & -2 & 6 & 1 \\ 7 & 0 & 8 & -3 \\ \sqrt{2} & \frac{3}{5} & 4 & 3\end{array}\right]$ then write
i. the elements $a_{23}, a_{31}, a_{14}, a_{33}, a_{22}$ of $A$.
Solution: (i) $a_{i j}=$ element of $t^{\text {th }}$ row and $j^{\text {th }}$ column $\begin{array}{l} a_{23}=8 \\ a_{31}=\sqrt{2} \\ a_{14}=1 \\ a_{33}=4 \\ a_{22}=0 \end{array}$
If $A=\left[\begin{array}{cccc}5 & -2 & 6 & 1 \\ 7 & 0 & 8 & -3 \\ \sqrt{2} & \frac{3}{5} & 4 & 3\end{array}\right]$ then write
i. the order of the matrix $A$,
ii. the number of all entries in $A$,
Solution: (i) Order of matrix $=$ Number of rows $x$ Number of columns $=(3 \times 4)$ (ii) Number of entries = (Number of rows) $x$ (Number of columns) $=3 \times 4$ $=12$
If $A=\left[\begin{array}{cccc}5 & -2 & 6 & 1 \\ 7 & 0 & 8 & -3 \\ \sqrt{2} & \frac{3}{5} & 4 & 3\end{array}\right]$ then write
i. the number of rows in $A$,
ii. the number of columns in $A$,
Solution: (i) Number of rows $=3$ (ii) Number of columns = 4