First slide

Operations of matrix

Question

Let $A = (\begin{array}{ll} a ââââ b \\ c ââââ d \end{array})$ be such that $A^{3} = O$ but $A â O$ , then

Moderate

A

$A^{2} = O$
B

$A^{2} = A$
C

$A^{2} = I â A$
D

none of these

Solution

As $A^{3} = O$ , we get $|A^{3}| = 0$
$| A |^{3} = 0 â | A | â a d â b c = 0$
Also,
$\begin{array}{l} A^{2} = (\begin{array}{cc} a^{2} + b c & (a + d) b \\ (a + d) c & b c + d^{2} \end{array}) \\ = (\begin{array}{ll} a^{2} + a d & (a + d) b \\ (a + d) c & a d + d^{2} \end{array}) \\ = (a + d) A \end{array}$
If $a + d = 0$ we get $A^{2} = O$ .
Suppose $a + d â 0$ , then
$\begin{array}{l} A^{3} = A^{2} A = (a + d) A^{2} \\ â O = (a + d) A^{2} â A^{2} = O \end{array}$

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