If A = 2 - 3 - 4 1 , then A T 2 + 12 A T =

Given:

$A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]$

$\Rightarrow A^T=\left[\begin{array}{cc}2& -4\\ -3& 1\end{array}\right]$

$\Rightarrow {\left(A^T\right)}^2=\left[\begin{array}{cc}2& -4\\ -3& 1\end{array}\right]\left[\begin{array}{cc}2& -4\\ -3& 1\end{array}\right]$

$\Rightarrow {\left(A^T\right)}^2=\left[\begin{array}{cc}16& -12\\ -9& 13\end{array}\right]$

And,

$12A=\left[\begin{array}{cc}24& -36\\ -48& 12\end{array}\right]$

$\Rightarrow {\left(12A\right)}^T=\left[\begin{array}{cc}24& -48\\ -36& 12\end{array}\right]$

So,

${\left(A^T\right)}^2+{\left(12A\right)}^T=\left[\begin{array}{cc}16& -12\\ -9& 13\end{array}\right]+\left[\begin{array}{cc}24& -48\\ -36& 12\end{array}\right]$

$\Rightarrow {\left(A^T\right)}^2+{\left(12A\right)}^T=\left[\begin{array}{cc}40& -60\\ -45& 25\end{array}\right]$