Question

Find the inverse of each matrix, if it exists.27.

$\left[\begin{array}{cc}-2& 0\\ 5& 6\end{array}\right]$

Short answer

The inverse of the matrix is$\left[\begin{array}{cc}-\frac{1}{2}& 0\\ \frac{5}{12}& \frac{1}{6}\end{array}\right]$

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$\left[\begin{array}{cc}-2& 0\\ 5& 6\end{array}\right]$

That is,$A=\left[\begin{array}{cc}-2& 0\\ 5& 6\end{array}\right]$

Comparing with the standard form $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$,$a=-2,b=0,c=5,d=6$ .Then, $A^{-1}$is:

$\begin{array}{c}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ =\frac{1}{(-2\times 6)-(0\times 5)}\left[\begin{array}{cc}6& 0\\ -5& -2\end{array}\right]\\ =\frac{1}{-12-0}\left[\begin{array}{cc}6& 0\\ -5& -2\end{array}\right]\\ =\frac{1}{-12}\left[\begin{array}{cc}6& 0\\ -5& -2\end{array}\right]\end{array}$

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

As $ad-bc\ne 0$here, inverse exist.

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

$\begin{array}{c}{A}^{-1}=\frac{1}{-12}\left[\begin{array}{cc}6& 0\\ -5& -2\end{array}\right]\\ =\left[\begin{array}{cc}\frac{6}{-12}& \frac{0}{-12}\\ \frac{-5}{-12}& \frac{-2}{-12}\end{array}\right]\\ =\left[\begin{array}{cc}-\frac{1}{2}& 0\\ \frac{5}{12}& \frac{1}{6}\end{array}\right]\end{array}$

- State the conclusion.

Therefore, the inverse of the matrix$\left[\begin{array}{cc}-2& 0\\ 5& 6\end{array}\right]$is$\left[\begin{array}{cc}-\frac{1}{2}& 0\\ \frac{5}{12}& \frac{1}{6}\end{array}\right]$