📘 Question
If \(A\) and \(B\) are square matrices of order 3 such that:
\[
AB = O
\]
and \(A\) is non-singular, then \(B\) is:
(a) null matrix
(b) singular matrix
(c) unit matrix
(d) non-singular matrix
✏️ Step-by-Step Solution
Step 1: Use inverse of non-singular matrix
Since \(A\) is non-singular, its inverse \(A^{-1}\) exists.
Step 2: Multiply both sides by \(A^{-1}\)
\[
A^{-1}(AB) = A^{-1}O
\]
Step 3: Simplify
\[
(A^{-1}A)B = O
\]
\[
IB = O
\Rightarrow B = O
\]
✅ Final Answer
\[
\boxed{(a)\; \text{null matrix}}
\]
💡 Key Concept
If a matrix is non-singular (invertible), it cannot produce zero unless multiplied by the zero matrix.
\[
AB = O \;\text{and}\; A^{-1} \text{ exists} \Rightarrow B = O
\]