Question

$IfA=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$, then the determinant $A^2-2A$:

• A. $5$
• B. $25$
• C. $-5$
• D. $-25$

Answer 1

The correct option is B $25$

Given $A=\left(\begin{array}{cc}1& 3\\ 2& 1\end{array}\right)$
$A^2=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$
By operation of matrix (4),
$A^2=\left[\begin{array}{cc}7& 6\\ 4& 7\end{array}\right]$
So,$A^2-2A=\left[\begin{array}{cc}7& 6\\ 4& 7\end{array}\right]-2\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}7& 6\\ 4& 7\end{array}\right]-\left[\begin{array}{cc}2& 6\\ 4& 2\end{array}\right]$
$=\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$
So, $det{A}^2-2A=|A^2-2A|=25$