Q24. - Algebra 2

Question

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

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

Step-by-Step Solution

Answer

The inverse of the matrix does not exist.

Step 1 - 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.

Step 2 - 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.

Step 3 - State the conclusion.

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