First slide

Operations of matrix

Question

If A, B, A + I, A + B are idempotent matrices, then AB is equal to

Moderate

A

BA
B

-BA
C

I
D

O

Solution

Given A, B, A + I, A + B are idempotent. Hence,
$\begin{array}{l} A^{2} = A, B^{2} = B, (A + I)^{2} = A + I and (A + B)^{2} = A + B \\ \Rightarrow A^{2} + B^{2} + AB + BA = A + B \\ \Rightarrow A + B + AB + BA = A + B \\ \Rightarrow AB + BA = O \end{array}$

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