Let `Aa n dB` be two `2xx2` matrices. Consider the statements
(i) `A B=O, A=O or B=O`
(ii) `A B=I_2 implies A=B^(-1)`
(iii) `(A+B)^2=A^2+2A B+B^2`
a. (i) and (ii) are false, (iii) is true
b. (ii) and (iii) are false, (i) is true
c. (i) is false (ii) and, (iii) are true
d. (i) and (iii) are false, (ii) is true

Options

A. (i) and (ii) are false, (iii) is true
B. (ii) and (iii) are false, (i) is true
C. (i) is false, (ii) and (iii) are true
D. (i) and (iii) are false, (ii) is true

Answer

D

Explanation

(i) is false.
If `A=[(0,1),(0,-1)]` and `B=[(1,1),(0,0)]`, then `AB=[(0,0),(0,0)]=O`
(ii) is true as the product AB is an identity matrix, if and only if B is inverse of the matrix A.
(iii) is false since matrix multiplication in not commutative.