First slide

Operations of matrix

Question

If $A$ is $2 \times 2$ matrix such that $A^{2}$ = O, then $t r (A)$ is

Moderate

A

$1$
B

$0$
C

$- 1$
D

none of these

Solution

Let $A = [\begin{array}{ll} a b \\ c d \end{array}]$ , then $A^{2} = [\begin{array}{ll} a^{2} + b c a b + b d \\ a c + c d b c + d^{2} \end{array}]$
Now, $A^{2} = O$
$\begin{aligned} \Rightarrow a^{2} + b c = 0, (a + d) b = 0 \\ (a + d) c = 0, b c + d^{2} = 0 \end{aligned}$
Suppose $a + d \neq 0,$ then $b = 0, c = 0$
$∴ a^{2} = 0, d^{2} = 0 \Rightarrow a = 0, d = 0$
Contradiction.
Thus, $a + d = 0$ we get $t r (A) = 0$

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