Let $A$ be a $2 imes 2$ matrix with real entries. Let $I$ be the $2 imes 2$ identity matrix. $\operatorname{Tr}(A)$ denotes the sum of diagonal entries of $A$. Assume that $A^2=I$ Statement I If $A
eq I$ and $A
eq-1$, then $\operatorname{det} A=-1$ Statement II If $A
eq I$ and $A
eq-1$…

Question

Let $A$ be a $2 	imes 2$ matrix with real entries. Let $I$ be the $2 	imes 2$ identity matrix. $\operatorname{Tr}(A)$ denotes the sum of diagonal entries of $A$. Assume that $A^2=I$ Statement I If $A 
eq I$ and $A 
eq-1$, then $\operatorname{det} A=-1$ Statement II If $A 
eq I$ and $A 
eq-1$, then $\operatorname{Tr} A 
eq 0$

Aicharya · Accepted Answer

Let $A=\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}ight]$ $A^2=\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}ight]\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}ight]=\left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}ight]=I$ Here, $A^2=I,|A|=-1$ and $A 
eq \pm I$ $\operatorname{Tr}(A)=0$ $	herefore$ Statement I is true. Statement II is false.

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix.…

Solution