Answer
$\left|A^{T}\right|=-74$, $\left|A^{2}\right|=5476$
$\left|A A^{T}\right|=5476 \quad, \quad|2 A|=-296 $
$\left|A^{-1}\right|=\frac{-1}{74}$
Work Step by Step
Given the matrix
$A=\left[\begin{array}{ll}-4 & 10 \\ 5 & 6 \end{array}\right]$
We have
$A^{T}=\left[\begin{array}{cc}-4 & 5 \\ 10 & 6\end{array}\right] $ and $2 A=\left[\begin{array}{cc}-8 & 20 \\ 10 & 12\end{array}\right]$
We also have
$|A|=-24-50=-74, \left|A^{T}\right|=-24-50=-74$ and $\left|A^{2}\right|=|A| |A|=74 * 74=5476$
We obtain
$\quad\left|A A^{T}\right|=|A|\left|A^{T}\right|=74 * 74=5476 \quad, \quad|2 A|=-(8*12)-(20*10)=-296 $
and $\left|A^{-1}\right|=\frac{1}{|A|}=\frac{-1}{74}$
