First slide

Operations of matrix

Question

If B is an idempotent matrix, and A = I - B, then

Moderate

A

A² = A
B

A² = I
C

AB = O
D

BA = O

Solution

B is an idempotent matrix
$∴ B^{2} = B$
Now, $A^{2} = (I - B)^{2}$
$\begin{array}{l} = (I - B) (I - B) \\ = I - IB - IB + B^{2} \\ = I - B - B + B^{2} \\ = I - 2 B + B^{2} \\ = I - 2 B + B \\ = I - B \\ = A \end{array}$
Therefore, A is idempotent. Again,
$AB = (I - B) B = IB - B^{2} = B - B^{2} = B^{2} - B^{2} = O$
Similarly, $BA = B (I - B) = BI - B^{2} = B - B = O$ .

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