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Question

Let A be a nilpotent matrix of index m. If (IA)n=I+A+A2++Am1, where I is the identity matrix of order same as matrix A, then the value of n is

A
m
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B
m1
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C
1
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D
1
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Solution

The correct option is C 1
A is nilpotent matrix of index m.
Am=O
Let B=I+A+A2++Am1
Post-multiplying with (IA) on both sides,
B(IA)=(I+A+A2++Am1)(IA)
B(IA)=I+A+A2++Am1(A+A2++Am1+Am)
B(IA)=IAm
B(IA)=I
B=(IA)1
n=1

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