Let $A$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $\mathrm{A}^{2}$ is 1 , then the possible number of such matrices is
Correct Option: 1
$A=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)$
$\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{I}$
$A^{2}=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)=\left(\begin{array}{cc}a^{2}+b^{2} & b(a+c) \\ b(a+c) & b^{2}+c^{2}\end{array}\right)$
Sum of the diagonal entries of
$\mathrm{A}^{2}=\mathrm{a}^{2}+2 \mathrm{~b}^{2}+\mathrm{c}^{2}$
Given $a^{2}+2 b^{2}+c^{2}=1, a, b, c \in I$
$b=0 \& a^{2}+c^{2}=1$
Case-1 : $a=0 \Rightarrow c=\pm 1$
(2-matrices)
Case-2 : $\mathrm{c}=0 \Rightarrow \mathrm{a}=\pm 1$
(2-matrices)
Total $=4$ matrices
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.