Question

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A^2$ is $1$, then the possible number of such matrices is

• A. 4
• B. 1
• C. 6
• D. 12

Answer 1

Answer: (A)

$A=\left(\begin{array}{cc}a& b\\ b& c\end{array}\right),a,b,c\in I$
$A^2=\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)=\left(\begin{array}{cc}{a}^2+b^2& b\left(a+c\right)\\ b\left(a+c\right)& b^2+c^2\end{array}\right)$
Sum of the diagonal entries of
$A^2=a^2+2b^2+c^2$
Given $a^2+2b^2+c^2=1,a,b,c\in I$
$b=0$ & $a^2+c^2=1$
Case-1 : $a=0\Rightarrow c=\pm 1$ (2-matrices)
Case-2 : $c=0\Rightarrow a=\pm 1$ (2-matrices)
Total $=4$ matrices