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Question

If A=[1234], then A45A3A24AI is equal to

A
zero matrix
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B
I
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C
A
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D
A+I
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Solution

The correct option is D A+I
We know that every square matrix satisfies its characteristic equation.
|AλI|=0
1λ234λ=0
λ25λ2=0
λ2=5λ+2
A2=5A+2I
A3=5A2+2A
A4=5A3+2A2
A45A3A2=A2
A45A3A2=5A+2I
A45A3A24AI=A+I

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