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If A=\begin{bmatrix} 2 &-3 &5 \\ 3 & 2& -4\\ 1 & 1&-2 \end{bmatrix} , then find A^{-1} .

Using A^{-1}, solve the following system of equations :

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = - 3

 

 
 
 
 
 

Answers (1)
S safeer

Given: 

A=\begin{bmatrix} 2 &-3 &5 \\ 3 & 2& -4\\ 1 & 1&-2 \end{bmatrix}

\\|A|=2(-4+4)+3(-6+4)+5(3-2)\\ =0-6+5=-1

Adj(A)=\left[\begin{array}{ccc} 0 & 2 & 1 \\ -1 & -9 & -5 \\ 2 & 23 & 13 \end{array}\right] ^T

Adj(A)=\left[\begin{array}{ccc} 0 & -1 & 2 \\ 2 & -9 & 23 \\ 1 & -5 & 13 \end{array}\right]

A^{-1}=\frac{Adj(A)}{|A|}

A^{-1}=\frac{1}{-1}\left[\begin{array}{ccc} 0 & -1 & 2 \\ 2 & -9 & 23 \\ 1 & -5 & 13 \end{array}\right] = \left[\begin{array}{ccc} 0 & 1 & -2 \\ -2 & 9 & -23 \\ -1 & 5 & -13 \end{array}\right]

Given Equation: 

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = - 3

\\ \text { Given equations can be written as } A X=B \\ \text { Where } X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right], B=\left[\begin{array}{c} 11 \\ -5 \\ -3 \end{array}\right]

\\X=A^{-1} B\\\\ X=\left[\begin{array}{ccc} 0 & 1 & -2 \\ -2 & 9 & -23 \\ -1 & 5 & -13 \end{array}\right]\left[\begin{array}{c} 11 \\ -5 \\ -3 \end{array}\right]\\ X= \left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]

x =1 , y = 2, z=3

 

 

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