The correct answer is Option (B). Explanation: $A A^{-1}=\mid$ $\operatorname{det}\left(A A^{-1}=I\right)$ $\operatorname{det}(A) \operatorname{det}\left(A^{-1}\right)=1$...
Let $A$ be a non-singular matrix of order $3 \times 3$. Then |adj. $A \mid$ is equal to: (A) $|A|$ (B) $|\mathbf{A}|^{2}$ (C) $|\mathbf{A}|^{3}$ (D) $3|\mathrm{~A}|$
The correct answer is Option (B). Explanation: $\mid$ adj. $\left.A|=| A\right|^{n-1}=|A|^{2} \quad($ for $n=3)$
If a=(fig 1) find $A^{3}-6 A^{2}+9 A-4 I=0$ and hence find A inverse.
fig(1) A = = Now = = L.H.S. = = = = = = = R.H.S. Now, to find , multiplying by =
For the matrix A = (FIG 1) show that $A^{3}-6 A^{2}+5 A+11 \mid$=0 Hence find A inverse.
fig(1) A = = $A^{3}=A^{2} A=\left[\begin{array}{ccc}4 & 2 & 1 \\ -3 & 8 & -14 \\ 7 & -3 & 14\end{array}\right]\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2...
For the matrix $A=\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right]$, find the numbers $a$ and $b$ such that $A^{2}+a A+b l=0$.
$A^{2}=A A=\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right]=\left[\begin{array}{ll}11 & 8 \\ 4 &...
If $A=\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right]$, show that $A^{2}-5 A+71=0$. Hence find $A^{-1}$.
$A^{2}=A A$ $A^{2}=\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right]=\left[\begin{array}{rr}8 & 5 \\ -5 &...
Let $A=\left[\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right]$ and $B=\left[\begin{array}{cc}6 & 8 \\ 7 & 9\end{array}\right]$ verify that $(A B)^{-1}=B^{-1} A^{-1}$
$A=\left[\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right]$ $|\mathrm{A}|=\left|\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right|=1 \neq 0$ $\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|}$...
Find the inverse of each of the matrices (if it exists) given in $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & \sin \alpha & -\cos \alpha\end{array}\right]$
Let A = = exists. A11 = , A12 = , A13 = , A21 = , A22 = , A23 = , A31 = , A32 = , A33 = adj. A =
Find the inverse of each of the matrices (if it exists) given in $\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]$
Let A = = exists. A11 = , A12 = , A13 = , A21 = , A22 = , A23 = , A31 = , A32 = , A33 = adj. A =
Find the inverse of each of the matrices (if it exists) given in $\left[\begin{array}{ccc}2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1\end{array}\right]$
Let $\mathrm{A}=\left[\begin{array}{ccc}2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1\end{array}\right]$ Therefore, $A^{-1}$ exists A11 = , A12 = , A13 = , A21 = , A22 = , A23 = ,...
Find the inverse of each of the matrices (if it exists) given in $\left[\begin{array}{ccc}1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1\end{array}\right]$
Let $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1\end{array}\right]$ Therefore, $A^{-1}$ exists Find adj A: 212. \text { 212. }...
Find the inverse of each of the matrices (if it exists) given in $\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5\end{array}\right]$
$\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5\end{array}\right]$ Solution: Let $\mathrm{A}=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0...
Find the inverse of each of the matrices (if it exists) given in $\left[\begin{array}{ll}-1 & 5 \\ -3 & 2\end{array}\right]$
Let $A=\left[ {{a}_{ij}} \right]$ be a square matrix of order n. The adjoint of a matrix A is the transporse of the cofactor matrix of A. It is denoted by adj A. Given...
Find the inverse of each of the matrices (if it exists) given in $\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right]$
Let $A=\left[\begin{array}{rr}2 & -2 \\ 4 & 3\end{array}\right]$ $|A|=\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right]=14 \neq 0$ Since determinant of the matrix is not zero,...
Verify $A(\operatorname{adj} \mathbf{A})=(\operatorname{adj} \mathbf{A}) \mathbf{A}=|\mathbf{A}| \mathbf{I}$ in $\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3\end{array}\right]$
. Let A = = adj. A = A. (adj. A) = = = ……….(i) Again (adj. A). A = = = ……….(ii) And = Also = ……….(iii) From eq. (i), (ii) and (iii) A. (adj. A) = (adj. A). A...
Verify $A(\operatorname{adj} \mathbf{A})=(\operatorname{adj} \mathbf{A}) \mathbf{A}=|\mathbf{A}| \mathbf{I}$ in $\left[\begin{array}{cc}2 & 3 \\ -4 & -6\end{array}\right]$
Let A = adj. A = A.(adj. A) = = = …..(i) Again (adj. A). A = = = …..(ii) And = Again …..(iii) From eq. (i), (ii) and (iii) A. (adj. A) = (adj. A). A...
Find adjoint of each of the matrices in $\left[\begin{array}{ccc}1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1\end{array}\right]$
Let $\mathrm{A}=\left[\begin{array}{ccc}1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1\end{array}\right]$ Cofactors of the above matrix are $\mathrm{A}_{11}=+\left|\begin{array}{ll}3...
Find adjoint of each of the matrices in $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$
Let $A=$ $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ Cofactors of the above matrix are 11 $A_{11}=4$ $A_{12}=-3$ $\mathrm{A}_{21}=-2$ $A_{22}=1$ adj....