Matrices & Determinants
Properties of matrices
Grade 12
Question:
<p>If <em>A</em>, <em>B</em>, <em>C</em> are nonsingular square matrices of order 3 × 3 then which of the following is not necessarily true?</p>
<p>(a) \(A^T A = I \Rightarrow AA^T = I\)</p>
<p>(b) \(A^{-1}B^{-1} = (BA)^{-1}\)</p>
<p>(c) \(I^{-1} = I\)</p>
<p>(d) \((A - B)C = AC - BC\)</p>
Step-by-Step Solution
Key Concept: Not all matrix operations preserve the cancellation laws that hold for real numbers. Specifically, matrix multiplication is not commutative (AB ≠ BA in general), so AB = AC does not necessarily imply B = C unless we know A is nonsingular and explicitly pre-multiply by A⁻¹.
<p><strong>Step 1:</strong> Examine each option involving properties of nonsingular matrices A, B, C.</p><p><strong>Step 2:</strong> Check if AB = AC necessarily implies B = C: Multiplying by A⁻¹ on the left: A⁻¹(AB) = A⁻¹(AC) ⟹ (A⁻¹A)B = (A⁻¹A)C ⟹ IB = IC ⟹ B = C. ✓ TRUE</p><p><strong>Step 3:</strong> Check if (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹: By properties of inverse, (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹. ✓ TRUE</p><p><strong>Step 4:</strong> Check if det(AB) = det(A)det(B): This is a fundamental property of determinants. ✓ TRUE</p><p><strong>Step 5:</strong> Check if AB = BA (commutativity): This is NOT necessarily true. Matrices generally do not commute. ✗ NOT NECESSARILY TRUE</p><p>∴ Answer: A (The statement that AB = BA is not necessarily true)</p>
Correct Answer: A