Question

If $A=\left[\begin{array}{cc}2& a\\ -1& -2\end{array}\right]$ is an involutory matrix, then which of the following is/are TRUE

• A. $a=1$
• B. $a=3$
• C. $|A|=-1$
• D. $A^2=A$

Answer 1

Answer: (B), (C)

$|A|=-1$

Given $A=\left[\begin{array}{cc}2& a\\ -1& -2\end{array}\right]$ is an involutory matrix.
For an involutory matrix, $A^2=I$
Here $A^2=\left[\begin{array}{cc}2& a\\ -1& -2\end{array}\right]\left[\begin{array}{cc}2& a\\ -1& -2\end{array}\right]$
$=\left[\begin{array}{cc}4-a& 0\\ 0& -a+4\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow a=3$
And,$|A|=-1$