Question

Find the inverse of each matrix, if it exists.21.

$\left[\begin{array}{cc}6& 3\\ 8& 4\end{array}\right]$

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 A be the matrix$\left[\begin{array}{cc}6& 3\\ 8& 4\end{array}\right]$

That is,$A=\left[\begin{array}{cc}6& 3\\ 8& 4\end{array}\right]$

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

$a=6,b=3,c=8,d=4$

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

$\begin{array}{c}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ =\frac{1}{(6\times 4)-(8\times 3)}\left[\begin{array}{cc}4& -3\\ -8& 6\end{array}\right]\\ =\frac{1}{24-24}\left[\begin{array}{cc}4& -3\\ -8& 6\end{array}\right]\end{array}$

Here,$\begin{array}{c}ad-bc=24-24\\ =0\end{array}$

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 $\left[\begin{array}{cc}6& 3\\ 8& 4\end{array}\right]$ does not exist.