Question

Let A be a $2 \times 2$ matrix with non-zero entries and let $A^2 = I$, where I is $2 \times 2$ identity matrix. Define Tr(A) = sum of diagonal elements of A and $|A|$ = determinant of matrix A. $Tr(A) = 0$ $|A| = 1$

• A. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1
• B. Statement-1 is true, Statement-2 is false
• C. Statement-1 is false, Statement-2 is true
• D. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Answer 1

Answer: (B)

Statement-1 is true, Statement-2 is false

Answer 2

Let $A = \begin{pmatrix}a&b\\\ c&d\end{pmatrix}, abcd \ne 0$ $A^{2} = \begin{pmatrix}a&b\\\ c&d\end{pmatrix}\cdot\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ $\Rightarrow A^{2} = \begin{pmatrix}a^{2}+bc&ab+bd\\\ ac+cd&bc+d^{2}\end{pmatrix}$ $\Rightarrow a^{2} + bc = 1, bc + d^{2} = 1$ $ab + bd = ac + cd = 0$ $c \ne 0$ and $b \ne 0\quad \Rightarrow a + d = 0$ Trace $A = a + d = 0$ $|A| = ad - bc = -a^{2} - bc = -1.$