Solution: Suppose the numbers as $x$, $y$, and $z$ $x+y+z=10,000 \ldots \ldots \text { (i) }$ Also, $0.1 \mathrm{x}+0.12 \mathrm{y}+0.15 z=1310 \ldots \ldots \text { (ii) }$ Again, $0.1 x+0.12...
The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.
Solution: Suppose the numbers as $x$, $y$, $z$ $\begin{array}{l} x+y+z=2 \\ \ldots \cdots(i) \end{array}$ Also, $2 y+(x+z)+1$ $x+2 y+z=1 \ldots \ldots \text { (ii) }$ Again, $\begin{array}{l}...
(i) Given $A=\left[\begin{array}{ccc}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]$ find $B A$ and $U$ se this to solve the system of linear equations $y+2 z=$ $7, x-y=3,2 x+3 y+4 z=17$
Solution: Given that $\begin{array}{l} B=\left[\begin{array}{ccc} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{array}\right]_{A}=\left[\begin{array}{ccc} 2 & 2 & -4 \\...
(i) $A=\left[\begin{array}{ccc}1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1\end{array}\right]$, and $B=\left[\begin{array}{ccc}7 & 2 & -6 \\ -2 & 1 & -3 \\ -4 & 2 & 5\end{array}\right]$ find $A B$. Hence solve the system of linear equations: $x-2 y=10,2 x+y+3 z=8$ and $-2 y+z=7$
(ii) If $A=\left[\begin{array}{ccc}1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1\end{array}\right]$, find $A^{-1} .$ Using $A^{-1}$ solve the system of linear equations: $x-2 y=10,2 x-y-z=8,-2 y+z=7$
(i) Solution: $\begin{array}{l} A=\left[\begin{array}{ccc} 1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1 \end{array}\right]_{B}=\left[\begin{array}{ccc} 7 & 2 & -6 \\ -2...
$(i)$ If $A=\left[\begin{array}{ccc}1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1\end{array}\right]$, find $A^{-1} .$ Using $A^{-1}$, solve these stem of linear equations : $x-2 y=10,2 x+y+3 z=8,-2 y+z=7$
(ii) $A=\left[\begin{array}{ccc}3 & -4 & 2 \\ 2 & 3 & 5 \\ 1 & 0 & 1\end{array}\right]$, find $A^{-1}$ and hence solve thesystem of linear equations: $3 x-4 y+2 z=-1,2 x+3 y+5 z=7, x+z=2$
(i) Solution: Given that $\begin{array}{l} A=\left[\begin{array}{ccc} 1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1 \end{array}\right] \\ |A|=1(1+6)+2(2-0)+0 \\ =7+4 \\ =11...
Find $A^{-2}$, if $A=\left[\begin{array}{ccc}1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1\end{array}\right] .$ Hence solve the following system of linear equations : $x+2 y+5 z=10, x-y-z=-2,2 x+3 y-z=-11$
Solution: Given that $\begin{array}{l} A=\left[\begin{array}{ccc} 1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1 \end{array}\right] \\ |\mathrm{A}|=1(1+3)+2(-1+2)+5(3+2) \\ =4+2+25...
$\left[\begin{array}{ccc}2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2\end{array}\right]$, find $A^{-1}$ and hence solve the system of linear equations : $2 x-3 y+5 z=11,3 x+2 y-4 z=-5, x+y-2 z=-3$
Solution: $\begin{array}{r} A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{array}\right] \\ |\mathrm{A}|=2(0)+3(-2)+5(1) \\ =-1 \end{array}$ Now, the...
$I f A=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & 1 & -5\end{array}\right]$ are two square matrices. Find $A B$ and hence solve the system of linear equations : $x-y=3,2 x+3 y+4 z=17, y+2 z=7$
Solution: $\begin{array}{l} A=\left[\begin{array}{ccc} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{array}\right]_{B}=\left[\begin{array}{ccc} 2 & 2 & -4 \\ -4 & 2...
Show that each one of the following systems of linear equations is consistent:
(i) 3x – y – 2z = 2
2y – z = -1
3x – 5y = 3
(ii) x + y – 2z = 5
x – 2y + z = -2
-2x + y + z = 4
Solution: (i) Given that $3 x-y-2 z=2$ $\begin{array}{l} 2 y-z=-1 \\ 3 x-5 y=3 \\ {\left[\begin{array}{ccc} 3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0...
Show that each one of the following systems of linear equations is consistent:
(i) 4x – 2y = 3
6x – 3y = 5
(ii) 4x – 5y – 2z = 2
5x – 4y + 2z = -2
2x + 2y + 8z = -1
Solution: (i) Given that $4 x-2 y=3$ $6 x-3 y=5$ We can write the above system of equations as $\left[\begin{array}{ll} 4 & -2 \\ 6 & -3 \end{array}\right]\left[\begin{array}{l} \mathrm{X}...
Show that each one of the following systems of linear equations is consistent:
(i) 2x + 5y = 7
6x + 15y = 13
(ii) 2x + 3y = 5
6x + 9y = 10
Solution: (i) Given that $2 x+5 y=7$ $6 x+15 y=13$ We can write the above system of equations as $\left[\begin{array}{cc} 2 & 5 \\ 6 & 15 \end{array}\right]\left[\begin{array}{l} \mathrm{X}...
Show that each one of the following systems of linear equations is consistent and also find their solutions:
(i) 2x + 2y – 2z = 1
4x + 4y – z = 2
6x + 6y + 2z = 3
Solution: (i) Given that $x+y+z=6$ $\begin{array}{l} x+2 y+3 z=14 \\ x+4 y+7 z=30 \end{array}$ We can write this as $\begin{array}{l} {\left[\begin{array}{ccc} 2 & 2 & -2 \\ 4 & 4 &...
Show that each one of the following systems of linear equations is consistent and also find their solutions:
(i) 5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5
(ii) x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30
Solution: (i) Given that$5 x+3 y+7 z=4$ $\begin{array}{l} 3 x+26 y+2 z=9 \\ 7 x+2 y+10 z=5 \end{array}$ We can write this as: $\begin{array}{l} {\left[\begin{array}{ccc} 5 & 3 & 7 \\ 3 &...
Show that each one of the following systems of linear equations is consistent and also find their solutions:
(i) 6x + 4y = 2
9x + 6y = 3
(ii) 2x + 3y = 5
6x + 9y = 15
Solution: (i) Given that $6 x+4 y=2$ $9 x+6 y=3$ We can write the above system of equations as $\left[\begin{array}{ll} 6 & 4 \\ 9 & 6 \end{array}\right]\left[\begin{array}{l} \mathrm{X} \\...
Solve the following system of equations by matrix method:
(i) (2/x) + (3/y) + (10/z) = 4,
(4/x) – (6/y) + (5/z) = 1,
(6/x) + (9/y) – (20/z) = 2, x, y, z ≠ 0
(ii) x – y + 2z = 7
3x + 4y – 5z = -5
2x – y + 3z = 12
Solution: (i) Given that $(2 / x)+(3 / y)+(10 / z)=4$ $\begin{array}{l} (4 / x)-(6 / y)+(5 / z)=1 \\ (6 / x)+(9 / y)-(20 / z)=2, x, y, z \neq 0 \end{array}$ We can write the given system in matrix...
Solve the following system of equations by matrix method:
(i) 8x + 4y + 3z = 18
2x + y + z = 5
x + 2y + z = 5
(ii) x + y + z = 6
x + 2z = 7
3x + y + z = 12
Solution: (i) Given that $8 x+4 y+3 z=18$ $\begin{array}{l} 2 x+y+z=5 \\ x+2 y+z=5 \end{array}$ We can write the given system in matrix form as: $\begin{array}{l} {\left[\begin{array}{lll} 8 & 4...
Solve the following system of equations by matrix method:
(i) 2x + 6y = 2
3x – z = -8
2x – y + z = -3
(ii) 2y – z = 1
x – y + z = 2
2x – y = 0
Solution: (i) Given that $2 x+6 y=2$ $\begin{array}{l} 3 x-z=-8 \\ 2 x-y+z=-3 \end{array}$ We can write the given system in matrix form as: $\begin{array}{c} {\left[\begin{array}{ccc} 2 & 6...
Solve the following system of equations by matrix method:
(i) 3x + 4y + 2z = 8
2y – 3z = 3
x – 2y + 6z = -2
(ii) 2x + y + z = 2
x + 3y – z = 5
3x + y – 2z = 6
Solution: (i) Given that $3 x+4 y+2 z=8$ $\begin{array}{l} 2 y-3 z=3 \\ x-2 y+6 z=-2 \end{array}$ We can write the given system in matrix form as: $\left[\begin{array}{rrr} 2 & 1 & 1 \\ 1...
Solve the following system of equations by matrix method:
(i) (2/x) – (3/y) + (3/z) = 10
(1/x) + (1/y) + (1/z) = 10
(3/x) – (1/y) + (2/z) = 13
(ii) 5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25
Solution: (i) Given that $(2 / x)-(3 / y)+(3 / z)=10$ $\begin{array}{l} (1 / x)+(1 / y)+(1 / z)=10 \\ (3 / x)-(1 / y)+(2 / z)=13 \end{array}$ We can write the given system in matrix form as:...
Solve the following system of equations by matrix method:
(i) 6x – 12y + 25z = 4
4x + 15y – 20z = 3
2x + 18y + 15z = 10
(ii) 3x + 4y + 7z = 14
2x – y + 3z = 4
x + 2y – 3z = 0
Solution: (i) Given that $6 x-12 y+25 z=4$ $\begin{array}{l} 4 x+15 y-20 z=3 \\ 2 x+18 y+15 z=10 \end{array}$ We can write the given system in matrix form as: $\left[\begin{array}{ccc}6 & -12...
Solve the following system of equations by matrix method:
(i) x + y –z = 3
2x + 3y + z = 10
3x – y – 7z = 1
(ii) x + y + z = 3
2x – y + z = -1
2x + y – 3z = -9
Solution: (i) Given that $x+y-z=3$ $\begin{array}{l} 2 x+3 y+z=10 \\ 3 x-y-7 z=1 \end{array}$ We can write the given system in matrix form as: $\begin{array}{c} {\left[\begin{array}{ccc} 1 & 1...
Solve the following system of equations by matrix method:
(i) 3x + 7y = 4
x + 2y = -1
(ii) 3x + y = 7
5x + 3y = 12
Solution: (i) Given that $3 x+7 y=4$ $x+2 y=-1$ We can write the above system of equations as $\left[\begin{array}{ll} 3 & 7 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} \mathrm{X} \\...
Solve the following system of equations by matrix method:
(i) 3x + 4y – 5 = 0
x – y + 3 = 0
(ii) 3x + y = 19
3x – y = 23
Solution: (i) Given that $3 x+4 y-5=0$ $x-y+3=0$ We can write the above system of equations as $\left[\begin{array}{cc} 3 & 4 \\ 1 & -1 \end{array}\right]\left[\begin{array}{l} \mathrm{X} \\...
Solve the following system of equations by matrix method:
(i) 5x + 7y + 2 = 0
4x + 6y + 3 = 0
(ii) 5x + 2y = 3
3x + 2y = 5
Solution: (i) Given that $5 x+7 y+2=0$ and $4 x+6 y+3=0$ We can write the above system of equations as $\left[\begin{array}{ll} 5 & 7 \\ 4 & 6 \end{array}\right]\left[\begin{array}{l}...