Let A and B be two $$2x2$$ matrices. Consider the $$statements(i)$$ $$AB=O$$⇒$$A=O$$ or $$B=O(ii)$$ $$AB=I2$$⇒$$A=B$$−$$1(iii)$$ $$(A+B)2=A2+2AB+B2Then$$

Correct option is 2(i) and (iii) are false, (ii) is true

Questions

Let A and B be two 2x2 matrices. Consider the statements
(i) $A B = O \Rightarrow A = O$ or $B = O$
(ii) $A B = I_{2} \Rightarrow A = B^{- 1}$
(iii) $(A + B)^{2} = A^{2} + 2 A B + B^{2}$
Then

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(i) is false, (ii) and (iii) are true

(i) and (iii) are false, (ii) is true

(i) and (ii) are false, (iii) is true

(ii) and (iii) are false, (i) is true

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detailed solution

Correct option is B

(i) is false. If A=0 10 −1 and B=1 10 0, thenAB=0 00 0=OThus, AB=O⇒A=O or B=O(iii) is false since matrix multiplication is not commutative.(ii) is true as product AB is an identity matrix, if and only if B is inverse of the matrix A.

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Let A and B be two 2x2 matrices. Consider the statements(i) AB=O⇒A=O or B=O(ii) AB=I2⇒A=B−1(iii) (A+B)2=A2+2AB+B2Then

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Let A and B be two 2x2 matrices. Consider the statements
(i) $A B = O \Rightarrow A = O$ or $B = O$
(ii) $A B = I_{2} \Rightarrow A = B^{- 1}$
(iii) $(A + B)^{2} = A^{2} + 2 A B + B^{2}$
Then