Question

If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]$, then adj $(3A^2+12A)$ is equal to

• A. $\left[\begin{array}{cc}72& -84\\ -63& 51\end{array}\right]$
• B. $\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right]$
• C. $\left[\begin{array}{cc}51& 84\\ 63& 72\end{array}\right]$
• D. $\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right]$

$\begin{array}{c}A=\left(\begin{array}{cc}2& -3\\ -4& 1\end{array}\right)\\ A^2=\left(\begin{array}{cc}2& -3\\ -4& 1\end{array}\right)\left(\begin{array}{cc}2& -3\\ -4& 1\end{array}\right)\\ =\left(\begin{array}{cc}2\times 2-3\times -4& 2\times -3-3\times 1\\ -4\times 2+1\times -4& -4\times -3+1\times 1\end{array}\right)\\ =\left(\begin{array}{cc}16& -9\\ -12& 13\end{array}\right)\\ 3A^2=\left(\begin{array}{cc}48& -27\\ -36& 39\end{array}\right)\\ 12A=\left(\begin{array}{cc}24& -36\\ -48& 12\end{array}\right)\\ 3A^2+12A=\left(\begin{array}{cc}48+24& -27-36\\ -36-48& 39+12\end{array}\right)\\ =\left(\begin{array}{cc}72& -63\\ -84& 51\end{array}\right)\\ Co-factor\left(\begin{array}{cc}72& -63\\ -84& 51\end{array}\right)=\left(\begin{array}{cc}51& 84\\ 63& 72\end{array}\right)\\ adj\left(3A^2+12A\right)=\left(\begin{array}{cc}51& 63\\ 84& 72\end{array}\right)\\ =\left(\begin{array}{cc}51& 63\\ 84& 72\end{array}\right)\end{array}$