<p>Let <em>A</em> and <em>B</em> be matrices of order \(3 \times m\) and \(n \times 3\) respectively. If \(A + B\) as well as \(AB\) exist then</p>
Step-by-Step Solution
Key Concept: Matrix addition requires equal dimensions, while matrix multiplication AB requires A's column count to equal B's row count. These simultaneous constraints force A and B to be square matrices of order 3×3.
<p><strong>Step 1:</strong> For A+B to exist, matrices must have equal dimensions.</p><p>A is 3×m and B is n×3, so we need: 3=n and m=3</p><p>Therefore: A is 3×3 and B is 3×3</p><p><strong>Step 2:</strong> For AB to exist, columns of A must equal rows of B.</p><p>A is 3×m and B is n×3, so we need: m=n</p><p><strong>Step 3:</strong> Combining both conditions: m=3 and n=3 and m=n</p><p>This confirms A and B are both 3×3 matrices.</p><p>∴ Answer: m=n=3 (Both A and B are square matrices of order 3)</p>
Correct Answer: B