Question

Find the inverse of each matrix, if it exists.24.

$\left[\begin{array}{cc}-3& -2\\ 6& 4\end{array}\right]$

Short answer

The inverse of the matrix does not exist.

Step-by-step solution

- Define inverse of a matrix.

For the matrix A,

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

The inverse of matrix of the matrix A is:

$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

where $ad-bc\ne 0$. $ad-bc$ is the determinant of the matrix A

If $ad-bc=0$, the inverse of a matrix doesn not exist.

- Calculate the inverse.

Let A be the matrix $\left[\begin{array}{cc}-3& -2\\ 6& 4\end{array}\right]$

That is,$A=\left[\begin{array}{cc}-3& -2\\ 6& 4\end{array}\right]$

Comparing with the standard form $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, $a=-3,b=-2,c=6,d=4$.

Then, $A^{-1}$ is:$\begin{array}{c}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ =\frac{1}{(4\times -3)-(-2\times 6)}\left[\begin{array}{cc}4& -(-2)\\ -6& -3\end{array}\right]\\ =\frac{1}{-12+12}\left[\begin{array}{cc}4& 2\\ -6& -3\end{array}\right]\end{array}$

Here,

$\begin{array}{c}ad-bc=-12+12\\ =0\end{array}$

As $ad-bc=0$ here, inverse does not exist.

- State the conclusion.

Therefore, the inverse of the matrix $\left[\begin{array}{cc}-3& -2\\ 6& 4\end{array}\right]$ does not exist.