if `A` and `B` are two matrices of order `3xx3 `so that `AB=A` and `BA=B` then `(A+B)^7=`
if `A` and `B` are two matrices of order `3xx3 `so that `AB=A` and `BA=B` then `(A+B)^7=`
A. `7(A+B)`
B. `7. I_(3xx3)`
C. `64 (A+B)`
D. `128 I`
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Correct Answer - C
`AB=A, BA=B`
`ABA=A^(2)` or `A(BA)=A^(2)` or `A=A^(2)`
Similarly, `B^(2)=B`
`(A+B)^(2)=A^(2)+B^(2)+AB+BA`
`=A+B+A+B=2(A+B)`
`(A+B)^(3)=(A+B)^(2) (A+B)=2(A+B)^(2)=2^(2) (A+B)`
`implies (A+B)^(7)=2^(6)(A+B)=64 (A+B)`
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