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. $6$
• B. $1$
• C. $4$
• D. $12$

Answer 1

The correct option is C $4$

Let $A=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]=\left[\begin{array}{cc}{a}^2+b^2& ab+bc\\ ab+bc& c^2+b^2\end{array}\right]$

Given,sum of the diagonal entries of $A^2=1$
$\Rightarrow a^2+2b^2+c^2=1$
$\Rightarrow a=1,b=0,c=0$
or, $a=0,b=0,c=1$
or, $a=-1,b=0,c=0$
or, $a=0,b=0,c=-1$
Hence, possible number of such matrices is $4.$