Question

By using elementary operations, find the inverse of the matrix
$|1 - 1 \\ 2 \: 3|$
​

Answer 1

AnswEr :

Given Matrix,

$\left[\begin{array}{c c}1 & -1 \\2 & 3 \end{array} \right]$

The order of the matrix is 2 × 2,thus the identity matrix would be :

$\sf{I} = \left[\begin{array}{c c}1 & 0 \\ 0& 1 \end{array} \right]$

By Row Elementary Operations,

Suppose x is the inverse of the given matrix

$\implies \left[\begin{array}{c c}1 & -1 \\2 & 3 \end{array}\right] =\left[\begin{array}{c c}1 & 0 \\ 0& 1 \end{array}\right] \sf{x}$

• $\sf R_2 \longrightarrow R_2 - 2R_1$

Then,

$\implies \left[\begin{array}{c c}1 & -1 \\ 0 & 5 \end{array}\right] =\left[\begin{array}{c c}1 & 0 \\ -2 & 1 \end{array}\right] \sf{x}$

• $\sf R_2 \longrightarrow \dfrac{1}{5}R_2$

$\implies \left[\begin{array}{c c}1 & -1 \\ 0 & 1 \end{array}\right] =\left[\begin{array}{c c}1 & 0 \\ - \dfrac{2}{5}& \dfrac{1}{5} \end{array}\right] \sf{x}$

• $\sf R_1 \longrightarrow R_1 + R_2$

$\implies \left[\begin{array}{c c}1 & 0 \\ 0 & 1 \end{array}\right] =\left[\begin{array}{c c}\dfrac{3}{5} & \dfrac{1}{5} \\ \\ - \dfrac{2}{5}& \dfrac{1}{5} \end{array} \right] \sf{x}$

Inverse of the matrix would be :

$\left[\begin{array}{c c}\dfrac{3}{5} & \dfrac{1}{5} \\ \\ - \dfrac{2}{5}& \dfrac{1}{5} \end{array} \right]$