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Question

A and B are two non-singular square matrices of order 3×3 such that AB = A and BA = B then

A
|AB|=6
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B
|A+B|=0
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C
|AB|2=0
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D
|A+B|=8
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Solution

The correct options are
B |A+B|=0
C |AB|2=0
D |A+B|=8
AB=A,BA=B(A+B)2=A2+B2+AB+BAA2=AA=2(A+B)=ABA=AB=A|A+B|2=8|A+B|Similarly B2=B|A+B|(|A+B|8)=0 |A+B|=0 or |A+B|=8 Similarly|A+B|2=0

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