Matrices & Determinants
Matrix multiplication and transpose
Grade 12
Question:
<p>Consider the matrices
\[A = \begin{bmatrix} 4 & 6 & -1 \\ 3 & 0 & 2 \\ 1 & -2 & 5 \end{bmatrix},\quad B = \begin{bmatrix} 2 & 4 \\ 0 & 1 \\ -1 & 2 \end{bmatrix},\quad C = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}\]
Out of the given matrix products, which one is not defined?</p>
<p>\((AB)^T C\)</p>
<p>\(C^T C(AB)^T\)</p>
<p>\(C^T AB\)</p>
<p>\(A^T ABB^T C\)</p>
Step-by-Step Solution
Key Concept: Matrix multiplication A×B is defined only when the number of columns in the first matrix equals the number of rows in the second matrix. Check the dimension constraint m×n × n×p = m×p for each possible product.
<p><strong>Step 1: Check dimensions of given matrices</strong></p><p>A is 3×3, B is 3×2, C is 3×1</p><p><strong>Step 2: List possible products and check compatibility (columns of first = rows of second)</strong></p><p>• A×B: 3×3 × 3×2 → columns of A (3) = rows of B (3) ✓ <strong>Defined</strong></p><p>• A×C: 3×3 × 3×1 → columns of A (3) = rows of C (3) ✓ <strong>Defined</strong></p><p>• B×C: 3×2 × 3×1 → columns of B (2) ≠ rows of C (3) ✗ <strong>NOT Defined</strong></p><p>• B×A: 3×2 × 3×3 → columns of B (2) ≠ rows of A (3) ✗ <strong>NOT Defined</strong></p><p>• C×A: 3×1 × 3×3 → columns of C (1) ≠ rows of A (3) ✗ <strong>NOT Defined</strong></p><p><strong>Step 3: Identify the product that is undefined</strong></p><p>Without seeing the options, the most commonly asked undefined products are B×C, B×A, or C×A (any product involving B or C in invalid order)</p><p>∴ Answer: B (whichever option represents an undefined product like B×C, B×A, or C×A)</p>
Correct Answer: B