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Maths-

General

Easy

Question

Let A and B be two 2 × 2 matrices. Consider the statements
i) AB = O $rightwards double arrow$ A = O or B = O
ii) AB = I₂ $rightwards double arrow$ A = B^–1
iii) (A + B)² = A² + 2AB + B²
Then

(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

The correct answer is: (i) and (iii) are false, (ii) is true

(i) is false,
If A = $open square brackets table row 0 1 row 0 cell negative 1 end cell end table close square brackets$ and B = $open square brackets table row 1 1 row 0 0 end table close square brackets$ , then
AB = $open square brackets table row 0 0 row 0 0 end table close square brackets$ = O
Thus, AB = 0 $rightwards double arrow$ 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 B is inverse of the matrix A.

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