First slide

Applications of determinant

Question

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

Easy

A

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

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

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

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

Solution

(i) is false.
If $A = [\begin{array}{rr} 0 1 \\ 0 - 1 \end{array}]$ and $B = [\begin{array}{ll} 1 1 \\ 0 0 \end{array}]$ , then
$A B = [\begin{array}{ll} 0 0 \\ 0 0 \end{array}] = O$
Thus, $A B = O \Rightarrow A = O$ 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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