Question:
Let $\mathrm{A}$ 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: , 3
Solution:
Let $A=\left[\begin{array}{ll}a & b \\ b & c\end{array}\right]$
$A^{2}=\left[\begin{array}{ll}a^{L} & b \\ b & c\end{array}\right]\left[\begin{array}{ll}a & b \\ b & c\end{array}\right]=\left[\begin{array}{ll}a^{2}+b^{2} & a b+b c \\ a b+b c & c^{2}+b^{2}\end{array}\right]$
$=a^{2}+2 b^{2}+c^{2}=1$
$a=1, b=0, c=0$
$a=0, b=0, c=1$
$a=-1, b=0, c=0$
$c=-1, b=0, a=0$
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