## Precalculus (6th Edition) Blitzer

The provided matrix $A=\left[ \begin{matrix} 3 & 2 \\ 9 & 6 \\ \end{matrix} \right]$ is invertible; False.
Consider the matrix $A=\left[ \begin{matrix} 3 & 2 \\ 9 & 6 \\ \end{matrix} \right]$. Now, we will check if the matrix is invertible or not. Then, consider the matrix $A=\left[ \begin{matrix} 3 & 2 \\ 9 & 6 \\ \end{matrix} \right]$. Now, the inverse of matrix $\left[ A \right]$ is equal to: ${{\left[ A \right]}^{-1}}=\frac{1}{\left| ad-bc \right|}\left[ \begin{matrix} d & -b \\ -c & a \\ \end{matrix} \right]$ Now, compare the matrix to the original matrix. So, \begin{align} & a=3 \\ & b=2 \\ & c=9 \\ & d=6 \end{align} Now, the inverse is: ${{\left[ A \right]}^{-1}}=\frac{1}{\left| ad-bc \right|}\left[ \begin{matrix} d & -b \\ -c & a \\ \end{matrix} \right]$ Substitute the values to get, \begin{align} & {{\left[ A \right]}^{-1}}=\frac{1}{\left| ad-bc \right|}\left[ \begin{matrix} d & -b \\ -c & a \\ \end{matrix} \right] \\ & =\frac{1}{\left| 18-18 \right|}\left[ \begin{matrix} 6 & -2 \\ -9 & 3 \\ \end{matrix} \right] \\ & =\frac{1}{0}\left[ \begin{matrix} 6 & -2 \\ -9 & 3 \\ \end{matrix} \right] \end{align} So, $ad-bc=0$ which shows that the matrix is not invertible.