Question

Find the inverse of each matrix, if it exists.

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

Short answer

The inverse of the given matrix does not exist.

Step-by-step solution

Step 1. Given Information.

Given, to find the inverse of the matrix $\left[\begin{array}{cc}-3& 2\\ -6& 4\end{array}\right]$.

Step 2. Calculation.

The inverse of matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$is$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$, where$ad-bc\ne 0$:

Here$a=-3,b=2,c=-6,d=4$.

Checking if the inverse exists:

$\begin{array}{l}ad-bc=\left(-3\right)\left(4\right)-\left(2\right)\left(-6\right)\\ ad-bc=-12+12\\ ad-bc=0\end{array}$

Hence, the inverse does not exists.

Step 3. Conclusion.

Hence, the inverse of the given matrix does not exist.