Question

Let $A$ be a $2\times 2$ matrix with real entries, Let I be the $2\times 2$ identity matrix. Denote by $tr\left(A\right),$ the sum of diagonal entries of $A$. Assume that $A^2=I$
Statement $1:$ If $A\ne I$ and $A\ne -I$, then det$A=-1$
Statement $2:$ If $A\ne I$ and $A\ne -I$, then $tr\left(A\right)\ne 0,$ then which of the following is correct

• A. Statment $1$ is false, statement $2$ is true.
• B. Both statement $1$ & $2$ are true and statement $2$ is a correct explanation for statement $1$
• C. Both statement $1$ & $2$ are true but statement $2$ is not a correct explanation for statement $1$.
• D. Statement $1$ is true, statement $2$ is false

Answer 1

The correct option is D Statement $1$ is true, statement $2$ is false

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.
Then $A^2=\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+dc& bc+d^2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Hence
$\text{det}\left(A\right)=\left|\begin{array}{cc}\sqrt{1-bc}& b\\ c& -\sqrt{1-bc}\end{array}\right|=-1+bc-bc=-1$
Here, we can see $Tr\left(A\right)=0$ so, 2nd statement is false only.