Let $A$ be $a\,2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by tr$(A)$, 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 $(A)$ $ \ne 0$.

Question

Let $A$ be $a\,2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by tr$(A)$, 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 $(A)$ $ \ne 0$.

statement - 1 is false, statement -2 is true

statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1.

statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.

statement - 1 is true, statement - 2 is false.

Solution

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