First slide

Operations of matrix

Question

If Z is an idempotent matrix, then (I +Z)ⁿ

Moderate

A

I + 2ⁿZ
B

I + (2ⁿ - 1)Z
C

I - (2ⁿ - 1)Z
D

none of these

Solution

Z is idempotent, then
$\begin{matrix} Z^{2} = Z \Rightarrow Z^{3}, Z^{4}, \dots, Z^{n} = Z \\ ∴ (I + Z)^{n} =^{n} C_{0} I^{n} +^{n} C_{1} I^{n - 1} Z +^{n} C_{2} I^{n - 2} Z^{2} + +^{n} C_{n} Z^{n} \\ =^{n} C_{0} I +^{n} C_{1} Z +^{n} C_{2} Z +^{n} C_{3} Z + +^{n} C_{n} Z \\ = I + (^{n} C_{1} +^{n} C_{2} +^{n} C_{3} + +^{n} C_{n}) Z \\ = I + (2^{n} - 1) Z \end{matrix}$

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