Answer to Question #218163 in Linear Algebra for Dhruv rawat

Question

Find the inverse of the following matrix using row reduction method

[ 1 -1 2]
[ 3 2 4]
[ 0 1 -2]

Answer 1

To find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be the inverse matrix.

So, augment the matrix with the identity matrix:

$\begin{bmatrix} 1& -1 &2 & \kern{3 em} 1&0&0\\ 3& 2 &4 & \kern{3 em} 0&1&0\\ 0& 1 &-2 & \kern{3 em} 0&0&1 \end{bmatrix}$

$\rightarrow$ $R_2 = R_2 - 3R_1$

$\begin{bmatrix} 1& -1 &2 & \kern{3 em} 1&0&0\\ 0& 5 &-2 & \kern{3 em} -3&1&0\\ 0& 1 &-2 & \kern{3 em} 0&0&1 \end{bmatrix}$

$\rightarrow$ $R_2 = \dfrac{R_2 }{5}$

$\begin{bmatrix} 1& -1 &2 & \kern{3 em} 1&0&0\\ 0& 1 &-2/5 & \kern{3 em} -3/5&1/5&0\\ 0& 1 &-2 & \kern{3 em} 0&0&1 \end{bmatrix}$

$\rightarrow$ $R_1 = R_1 +R_2$

$\begin{bmatrix} 1& 0 &8/5 & \kern{3 em} 2/5 &1/5&0\\ 0& 1 &-2/5 & \kern{3 em} -3/5&1/5&0\\ 0& 1 &-2 & \kern{3 em} 0&0&1 \end{bmatrix}$

$\rightarrow$ $R_3 = R_3 - R_2$

$\begin{bmatrix} 1& 0 &8/5 & \kern{3 em} 2/5&1/5&0\\ 0& 1 &-2/5 & \kern{3 em} -3/5&1/5&0\\ 0& 0 &-8/5 & \kern{3 em} 3/5&-1/5&1 \end{bmatrix}$

$\rightarrow$ $R_3 = -\dfrac{5}{ 8}R_3$

$\begin{bmatrix} 1& 0 &8/5 & \kern{3 em} 2/5&1/5&0\\ 0& 1 &-2/5 & \kern{3 em} -3/5&1/5&0\\ 0& 0 &1 & \kern{3 em} -3/8&1/8&-5/8 \end{bmatrix}$

$\rightarrow$ $R_1 = R_1 - \dfrac{8}{5}R_3$

$\begin{bmatrix} 1& 0 &0 & \kern{3 em} 1&0&1\\ 0& 1 &-2/5 & \kern{3 em} -3/5&1/5&0\\ 0& 0 &1 & \kern{3 em} -3/8&1/8&-5/8 \end{bmatrix}$

$\rightarrow$ $R_2 = R_2 + \dfrac{2}{5}R_3$

$\begin{bmatrix} 1& 0 &0 & \kern{3 em} 1&0&1\\ 0& 1 &0 & \kern{3 em} -3/4&1/4&-1/4\\ 0& 0 &1 & \kern{3 em} -3/8&1/8&-5/8 \end{bmatrix}$

On the left is the identity matrix. On the right is the inverse matrix.

The inverse matrix is = $\begin{bmatrix} 1&0&1\\ -3/4&1/4&-1/4\\ -3/8&1/8&-5/8 \end{bmatrix}$