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
Select the correct option:
A
4
B
1
C
6
D
12
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
$$
\begin{aligned}
& A=\left(\begin{array}{ll}
a & b \\
b & c
\end{array}\right), \quad a, b, c \in I \\
& A^2=\left(\begin{array}{ll}
a & b \\
b & c
\end{array}\right)\left(\begin{array}{ll}
a & b \\
b & c
\end{array}\right)=\left(\begin{array}{ll}
a^2+b^2 & b(a+c) \\
b(a+c) & b^2+c^2
\end{array}\right)
\end{aligned}
$$
Sum of the diagonal entries of
$$
A^2=a^2+2 b^2+c^2
$$
Given $a^2+2 b^2+c^2=1, a, b, c \in I$
$$
b=0 \& a^2+c^2=1
$$
Case-1: $\mathrm{a}=0 \Rightarrow \mathrm{c}= \pm 1$ (2-matrices)
Case-2 : $\mathrm{c}=0 \Rightarrow \mathrm{a}= \pm 1$ (2-matrices)
Total = 4 matrices
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