Question

How to find the inverse of the matrix?

Answer 1

Step 1: Finding the determinant of matrix

Let consider the matrix $B=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$
Determinant of matrix $\left|B\right|=\left|\begin{array}{cc}1& 3\\ 2& 1\end{array}\right|$
$$\begin{gathered} =1-6 \\ =-5 \end{gathered}$$

If determinant is zero then inverse is not possible.

Step 2: Find the matrix of minors
Matrix of minors $=\left[\begin{array}{cc}1& 2\\ 3& 1\end{array}\right]$

With respect to $a_{11}$minor is $1$, $a_{12}$ minor is $2$ ,$a_{21}$ minor is $3$ and $a_{22}$minor is $1$

Step 3: Change into matrix of cofactors
Cofactors of matrix $=\left[\begin{array}{cc}1& -2\\ -3& 1\end{array}\right]$

Co-factor for ${\text{a}}_{ij}={(-1)}^{i+j}\times \text{minor}$

Step 4: Find the adjoint of matrix
Adjoint of matrix $Adj\left(A\right)={\left[\begin{array}{cc}1& -2\\ -3& 1\end{array}\right]}^T$
$=\left[\begin{array}{cc}1& -3\\ -2& 1\end{array}\right]$

Adjoint matrix is transpose of matrix formed by co-factors.

Step 5: Now multiply by the reciprocal of determinant
Inverse of matrix $B^{-1}=\frac{Adj\left(A\right)}{\left|B\right|}$
$$\begin{gathered} =\frac{1}{-5}\left[\begin{array}{cc}1& -3\\ -2& 1\end{array}\right] \\ =\left[\begin{array}{cc}-\frac{1}{5}& \frac{3}{5}\\ \frac{2}{5}& -\frac{1}{5}\end{array}\right] \end{gathered}$$

Hence above is the steps to find inverse of matrix