Practicing Success
Let A be a 2 × 2 matrix with non-zero entries and let $A^2=I$ where I is 2 × 2 identity matrix. Statement-1 $tr (A) = 0$ Statement-2 $|A|=1$ |
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. Statement-1 is False, Statement-2 is True. |
Statement-1 is True, Statement-2 is False. |
Let $A =\begin{bmatrix}a&b\\c&d\end{bmatrix}, a, b, c, d ≠0$. Then, $A^2=\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}a^2+bc&ab+bd\\ac+dc&bc+d^2\end{bmatrix}$ $∴A^2 = I$ $⇒a^2 +bc=1, bc+d^2=1, ac+dc=0$ and $ab+bd=0$ $⇒(a^2+bc) -(d^2 + bc) = 0, c (a + d) = 0$ and $(a + d) b=0$ $⇒a^2-d^2 =0$ and $a + d=0$ $[∵ b≠0, c ≠ 0]$ $⇒a+d=0⇒ Tr (A) = 0$ So, statement-1 is true. $|A|=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc=-a^2-bc$ $[∵d=-a]$ $⇒|A|=-1$ $[∵a^2+bc=1]$ So, statement-2 is not true. |