Question

If $A$ is a non-null diagonal matrix of order $3$ such that $A^4=A^2,$ then the possible number of matrices $A$ are

• A. $27$
• B. $26$
• C. $8$
• D. $7$

Answer 1

Answer: (B)

Let, $A=\left[\begin{array}{ccc}{d}_1& 0& 0\\ 0& d_2& 0\\ 0& 0& d_3\end{array}\right]$
$A^2=\left[\begin{array}{ccc}{d}_1^2& 0& 0\\ 0& d_2^2& 0\\ 0& 0& d_3^2\end{array}\right]\&A^4=\left[\begin{array}{ccc}{d}_1^4& 0& 0\\ 0& d_2^4& 0\\ 0& 0& d_3^4\end{array}\right]$
$\because A^4=A^2\Rightarrow d_1^4=d_1^2\Rightarrow d_1=0,1,-1$
$d_2^4=d_2^2\Rightarrow d_2=0,1,-1$
$d_3^4=d_3^2\Rightarrow d_3=0,1,-1$
$\because$ $A$ can't be a null matrix so the total number of possible matrices is $=3^3-1=26$