AIEEE 2008 (Offline)Question 1
Let $A$ be a $2\times2$ matrix with real entries. Let $I$ be the $2\times2$ identity matrix. Denote by $\operatorname{tr}(A)$, the sum of diagonal entries of $A$. Assume that $A^{2}=I$.
Statement-1: If $A\neq I$ and $A\neq-I$, then $\det(A)=-1$.
Statement-2: If $A\neq I$ and $A\neq-I$, then $\operatorname{tr}(A)\neq0$.
Matrices
Solution
$A^2=I$, $A\ne\pm I$ gives eigenvalues $1,-1$, so $\det A=-1$ (true) and $\operatorname{tr}A=0$, so Statement-2 is false.