Q46. - Algebra 2

Question

Which matrix does not have an inverse?

A  $\left[\begin{array}{cc}5& 3\\ 2& 4\end{array}\right]$ B$\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$  C  $\left[\begin{array}{cc}-3& -3\\ 6& -6\end{array}\right]$ D $\left[\begin{array}{cc}-10& -5\\ 8& 4\end{array}\right]$

Step-by-Step Solution

Answer

The matrix that does not have an inverse is:$D \left[\begin{array}{cc}-10& -5\\ 8& 4\end{array}\right]$

Step 1 ­- Description of step.

A square matrix $A$does not have its inverse if$\left|A\right|=0$.

A square matrix $A$ have its inverse if $\left|A\right|\ne 0$.

Step 2 ­- Find the determinant of the given matrices.

The determinant of the matrix $\left[\begin{array}{cc}5& 3\\ 2& 4\end{array}\right]$is:

$\begin{array}{c}\left|\begin{array}{cc}5& 3\\ 2& 4\end{array}\right|=\left(5\right)\left(4\right)-\left(2\right)\left(3\right)\\ =20-6\\ =14\end{array}$

The determinant of the matrix$\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$is:

$\begin{array}{c}\left|\begin{array}{cc}1& 2\\ 2& 1\end{array}\right|=\left(1\right)\left(1\right)-\left(2\right)\left(2\right)\\ =1-4\\ =-3\end{array}$

The determinant of the matrix $\left[\begin{array}{cc}-3& -3\\ 6& -6\end{array}\right]$is:

$\begin{array}{c}\left|\begin{array}{cc}-3& -3\\ 6& -6\end{array}\right|=\left(-3\right)\left(-6\right)-\left(6\right)\left(-3\right)\\ =18-\left(-18\right)\\ =18+18\\ =36\end{array}$

The determinant of the matrix $\left[\begin{array}{cc}-10& -5\\ 8& 4\end{array}\right]$is:

$\begin{array}{c}\left|\begin{array}{cc}-10& -5\\ 8& 4\end{array}\right|=\left(-10\right)\left(4\right)-\left(8\right)\left(-5\right)\\ =-40-\left(-40\right)\\ =-40+40\\ =0\end{array}$

As the determinant of the matrix

$\left[\begin{array}{cc}-10& -5\\ 8& 4\end{array}\right]$is equal to zero, therefore the inverse of the matrix $\left[\begin{array}{cc}-10& -5\\ 8& 4\end{array}\right]$does not exist. 

Step 3 ­- Description of step.

The matrix that does not have an inverse is$\left[\begin{array}{cc}-10& -5\\ 8& 4\end{array}\right]$.

Therefore, the option D is correct.