Q21. - Algebra 2

Question

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

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

Step-by-Step Solution

Answer

The inverse of the matrix is 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$\left[\begin{array}{cc}6& 3\\ 8& 4\end{array}\right]$

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

Step 3 - State the conclusion.

Therefore, the inverse of the matrix  $\left[\begin{array}{cc}6& 3\\ 8& 4\end{array}\right]$ does not exist.