<p>Let A be a 2 × 3 matrix whereas B be a 3 × 2 matrix. If det(AB) = 4, then find the value of det(BA).</p>
Step-by-Step Solution
Key Concept: The rank of a product matrix is limited by the dimensions of the factors; BA is 3×3 but has rank ≤ 2, so its determinant is 0
<p><strong>Solution:</strong> A is 2 × 3 and B is 3 × 2.</p><p>AB is a 2 × 2 matrix and BA is a 3 × 3 matrix.</p><p>Given det(AB) = 4.</p><p>Using the property: For rectangular matrices, if A is <i>m</i> × <i>n</i> and B is <i>n</i> × <i>m</i>, then:</p><p>$\text{det}(BA) - \text{det}(AB) = 0$ is <strong>not</strong> generally true.</p><p>However, rank(AB) ≤ min(2, 2) = 2 and rank(BA) ≤ min(3, 3) = 3.</p><p>The correct relationship is: $\text{det}(BA) = 0$ because BA is a 3 × 3 matrix with rank at most 2 (since A has only 2 rows).</p><p>∴ <strong>det(BA) = 0</strong></p>
Correct Answer: 0