Matrices & Determinants
Orthogonal Matrices
Grade 12
Question:
<p>If both \(A - \dfrac{I}{2}\) and \(A + \dfrac{I}{2}\) are orthogonal matrices, then which of the following statements are <strong>incorrect</strong>? (where \(I\) is an identity matrix of the same order as that of \(A\).)</p>
<p>\(A\) is skew-symmetric matrix of odd order.</p>
<p>\(A^2 = \dfrac{3}{4}I\)</p>
<p>\(A\) is skew-symmetric matrix of even order.</p>
<p>\(A\) is orthogonal</p>
Step-by-Step Solution
Key Concept: If both (A - I/2) and (A + I/2) are orthogonal, then (A - I/2)ᵀ(A - I/2) = I and (A + I/2)ᵀ(A + I/2) = I. Expanding these conditions simultaneously reveals constraints that lead to contradictions with standard matrix properties.
<p><strong>Step 1:</strong> If (A - I/2) is orthogonal, then (A - I/2)ᵀ(A - I/2) = I</p><p>Expanding: (Aᵀ - I/2)(A - I/2) = I</p><p>AᵀA - (Aᵀ + A)/2 + I/4 = I</p><p><strong>Step 2:</strong> If (A + I/2) is orthogonal, then (A + I/2)ᵀ(A + I/2) = I</p><p>Expanding: (Aᵀ + I/2)(A + I/2) = I</p><p>AᵀA + (Aᵀ + A)/2 + I/4 = I</p><p><strong>Step 3:</strong> Subtracting the first equation from the second:</p><p>(Aᵀ + A) = 0, which means A = -Aᵀ (A is skew-symmetric)</p><p><strong>Step 4:</strong> Substituting back into equation 1:</p><p>AᵀA - 0 + I/4 = I gives AᵀA = 3I/4</p><p><strong>Step 5:</strong> But for orthogonal A, we need AᵀA = I. Since AᵀA = 3I/4 ≠ I, matrix A cannot be orthogonal. Any statement claiming A is orthogonal or related properties would be incorrect.</p><p>∴ Answer: A</p>
Correct Answer: A