Question

Find the inverse of each matrix, if it exists.28.

$\left[\begin{array}{cc}-4& 6\\ 6& -9\end{array}\right]$

Short answer

The inverse of the matrix is 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 be the matrix$\left[\begin{array}{cc}-4& 6\\ 6& -9\end{array}\right]$

That is,$A=\left[\begin{array}{cc}-4& 6\\ 6& -9\end{array}\right]$

Comparing with the standard form $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, $a=-4,b=6,c=6,d=-9$.

Then, $A^{-1}$ is:

$\begin{array}{c}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ =\frac{1}{(-4\times -9)-(6\times 6)}\left[\begin{array}{cc}-9& -6\\ -6& -4\end{array}\right]\\ =\frac{1}{36-36}\left[\begin{array}{cc}-9& -6\\ -6& -4\end{array}\right]\end{array}$

Here, $\begin{array}{c}ad-bc=36-36\\ =0\end{array}$

And inverse exist only if $ad-bc\ne 0$.

As $ad-bc=0$ here, inverse doesnot exist.

- State the conclusion.

Therefore, the inverse of the matrix $\left[\begin{array}{cc}-4& 6\\ 6& -9\end{array}\right]$ does not exist.