If A and B are two square matrices of order 3 - 3 which satisfy AB - A and BA - B then A - B 7 isA- 7-13 - 3B- 12813 - 3C- 7 A - B D- 64 A - B

The correct option is D 64(A+B)AB =A, BA = B ⇒ A^2 = A and 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) ∴ ...

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Standard X

Mathematics

Multiplication of Matrices

If A and B ar...

Question

If A and B are two square matrices of order 3 $\times$ 3 which satisfy AB = A and BA = B then $(A + B)^{7}$ is

7 (A + B)

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{7.1}_{3 \times 3}

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64 (A + B)

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1281_{3 \times 3}

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Solution

The correct option is C $64 (A + B)$
AB =A, BA = B $\Rightarrow A^{2} = A$ and $B^{2} = B$
$(A + B)^{2} = A^{2} + B^{2} + A B + B A$
= A+B+A+B = 2(A+B)
$(A + B)^{3} = (A + B)^{2} (A + B) = 2 (A + B)^{2} = 2^{2} (A + B) ∴ (A + B)^{7} = 2^{6} (A + B) = 64 (A + B)$

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If A and B are two square matrices of order 3 × 3 which satisfy AB = A and BA = B then (A+B)7 is

The correct option is C 64(A+B)AB =A, BA = B ⇒A2=A and B2=B (A+B)2=A2+B2+AB+BA = A+B+A+B = 2(A+B) (A+B)3=(A+B)2(A+B)=2(A+B)2=22(A+B)∴(A+B)7=26(A+B)=64(A+B)

If A and B are two square matrices of order 3 $\times$ 3 which satisfy AB = A and BA = B then $(A + B)^{7}$ is

The correct option is C $64 (A + B)$
AB =A, BA = B $\Rightarrow A^{2} = A$ and $B^{2} = B$
$(A + B)^{2} = A^{2} + B^{2} + A B + B A$
= A+B+A+B = 2(A+B)
$(A + B)^{3} = (A + B)^{2} (A + B) = 2 (A + B)^{2} = 2^{2} (A + B) ∴ (A + B)^{7} = 2^{6} (A + B) = 64 (A + B)$