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Question

A,B,C are three square matrices of order n such that A=B+C and C is a nilpotent matrix with index 2. If B and C commute, then A10 is equal to


A
B10+12B9C  
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B
B10+10B9C  
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C
B10+12B10C   
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D
B10+10B10C   
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Solution

The correct option is B B10+10B9C  
C is nilpotent matrix with index 2.
C2=0
B and C commute.
BC=CB

A=B+CA2=(B+C)(B+C)=B2+BC+CB+C2=B2+2BC

A3=(B2+2BC)(B+C)=B3+B2C+2BCB+2BC2=B3+B2C+2B2C+O=B3+3B2C

A4=B4+4B3C

An=Bn+nBn1CA10=B10+10B9C

Mathematics

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