Matrices & Determinants
Orthogonal and Skew-Symmetric Matrices
Grade 12
Question:
<p>Given that both the matrices \(A - \dfrac{I}{2}\) and \(A + \dfrac{I}{2}\) are orthogonal, which of the following are correct?</p>
<p>(a) <em>A</em> is a skew-symmetric matrix</p>
<p>(b) <em>AA' = I</em></p>
<p>(c) <em>A</em> is an orthogonal matrix</p>
<p>(d) <em>A = −A'</em></p>
Step-by-Step Solution
Key Concept: If both A - I/2 and A + I/2 are orthogonal matrices, then their defining property (Q'Q = I) must hold for each. Using these two conditions simultaneously will constrain A to satisfy specific matrix properties.
<p><strong>Step 1: Apply orthogonality condition to A - I/2</strong></p><p>Since A - I/2 is orthogonal: (A - I/2)'(A - I/2) = I</p><p>Expanding: (A' - I/2)(A - I/2) = I</p><p>A'A - (I/2)A' - (I/2)A + I/4 = I</p><p>A'A - (I/2)(A' + A) + I/4 = I ... (1)</p><p><strong>Step 2: Apply orthogonality condition to A + I/2</strong></p><p>Since A + I/2 is orthogonal: (A + I/2)'(A + I/2) = I</p><p>Expanding: (A' + I/2)(A + I/2) = I</p><p>A'A + (I/2)A' + (I/2)A + I/4 = I</p><p>A'A + (I/2)(A' + A) + I/4 = I ... (2)</p><p><strong>Step 3: Subtract equation (1) from equation (2)</strong></p><p>[A'A + (I/2)(A' + A) + I/4] - [A'A - (I/2)(A' + A) + I/4] = 0</p><p>(I/2)(A' + A) + (I/2)(A' + A) = 0</p><p>(A' + A) = 0</p><p>∴ A' = -A</p><p><strong>Step 4: Verify each option</strong></p><p><strong>Option (a):</strong> A is skew-symmetric means A' = -A. ✓ From Step 3, this is TRUE.</p><p><strong>Option (b):</strong> AA' = I means A is orthogonal. From A' = -A, we get AA' = A(-A) = -A². For this to equal I, we'd need A² = -I. Let's check from the original equations: From equation (1): A'A - (I/2)(A' + A) + I/4 = I. Since A' + A = 0: A'A + I/4 = I, so A'A = 3I/4 ≠ I. ✗ FALSE.</p><p><strong>Option (c):</strong> A is orthogonal means AA' = I and A'A = I. From above, AA' = -A² and A'A = 3I/4 ≠ I. ✗ FALSE.</p><p><strong>Option (d):</strong> A = -A' means A' = -A, which is skew-symmetry. ✓ This is TRUE (equivalent to option a).</p><p><strong>∴ Answer:</strong> ABD</p>
Correct Answer: ABD