Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct? 

Explanation:

Given AB = O

Let  A = \(\left[\begin{array}{lll} a & b \\ c & d \end{array}\right]\)

B = \(\left[\begin{array}{lll} p & q \\ r & s \end{array}\right]\)

Now AB = \(\left[\begin{array}{lll} a & b \\ c & d \end{array}\right] \left[\begin{array}{lll} p & q \\ r & s \end{array}\right] = \left[\begin{array}{lll} 0 & 0 \\ 0 & 0 \end{array}\right]\)

\(\left[\begin{array}{lll} ap+br & aq+bs \\ cp+dr & cq+ds \end{array}\right]= \left[\begin{array}{lll} 0 & 0 \\ 0 & 0 \end{array}\right]\)

Comparing both sides, we get

⇒ ap + br = 0...(1)

⇒ aq + bs = 0...(2)

⇒ cp + dr = 0...(3)

⇒ cq + ds = 0...(4)

From Eqs. (i) and (iii), we get:

⇒ (ad – bc)p = 0 and (ad – bc)r = 0 

From Eqs. (ii) and (iv), we get:

⇒ (ad – bc) q = 0 and (ad – bc)s = 0 

Now, if A is non-singular

ad – bc ≠ 0

So p = q = r = s = 0.

Then,  B is the null matrix.

∴ Option (c) is correct.