First slide

Operations of matrix

Question

If $A^{2} = A$ , the $(I + A)^{4}$ equals

Moderate

A

$I + 15 A$
B

$I + 7 A$
C

$I + 8 A$
D

$I + 11 A$

Solution

As I and A commute, we can apply binomial theorem to expand ${(I + A)}^{4}$ . We have
$(I + A)^{4} = I + 4 A + 6 A^{2} + 4 A^{3} + A^{4}$ .
But $A^{3} = A^{2} = A, A^{4} = A^{3} = A$
$∴ (I + A)^{4} = I + 4 A + 6 A + 4 A + A = I + 15 A$

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