Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2=1$, where 1 is $2 \times 2$ identity…
Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2=1$, where 1 is $2 \times 2$ identity matrix. Define $\operatorname{Tr}(\mathrm{A})=$ sum of diagonal elements of $A$ and $|A|=$ determinant of matrix $A$.
Statement-1: $\operatorname{Tr}(\mathrm{A})=0$
Statement-2: $|\mathrm{A}|=1$
Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1
Statement-1 is true, Statement-2 is false
Statement-1 is false, Statement-2 is true
Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1
Solution
Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right), a b c d \neq 0$
$
\begin{aligned}
& A^2=\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right) \cdot\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right) \\
& \Rightarrow A^2=\left(\begin{array}{ll}
a^2+b c & a b+b d \\
a c+c d & b c+d^2
\end{array}\right) \\
& \Rightarrow a^2+b c=1, b c+d^2=1 \\
& a b+b d=a c+c d=0 \\
& c \neq 0 \text { and } b \neq 0 \\
& \text { Trace } A=a+d=0 \\
& |A|=a d-b c=-a^2-b c=-1 .
\end{aligned}
$