First slide

Operations of matrix

Question

If $A = [\begin{array}{cc} ab & b^{2} \\ - a^{2} & - ab \end{array}]$ then A is

Easy

A

Idempotent
B

Involutory
C

Nilpotent
D

Scalar

Solution

$A = [\begin{array}{cc} ab & b^{2} \\ - a^{2} & - ab \end{array}] A^{2} = [\begin{array}{cc} ab & b^{2} \\ - a^{2} & - ab \end{array}] [\begin{array}{cc} ab & b^{2} \\ - a^{2} & - ab \end{array}] = [\begin{array}{cc} a^{2} b^{2} - a^{2} b^{2} & a^{} b^{3} - a^{} b^{3} \\ - a^{3} b^{} + a^{3} b^{} & - a^{2} b^{2} + a^{2} b^{2} \end{array}] = [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]$

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