Matrices & Determinants
Operations on Matrices
Grade 12
Question:
<p>If \(A = \begin{pmatrix} a & b \\ b & a \end{pmatrix}\) and \(A^2 = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix}\), then</p>
<p>\(\alpha = a^2 + b^2,\, \beta = ab\)</p>
<p>\(\alpha = a^2 + b^2,\, \beta = 2ab\)</p>
<p>\(\alpha = a^2 + b^2,\, \beta = a^2 - b^2\)</p>
<p>\(\alpha = 2ab,\, \beta = a^2 + b^2\)</p>
Step-by-Step Solution
Key Concept: For a symmetric matrix of the form A = [[a,b],[b,a]], squaring it preserves the symmetric structure; compute A² directly and match coefficients to find α and β in terms of a and b.
<p><strong>Step 1:</strong> Compute A²:</p><p>A² = [[a,b],[b,a]] × [[a,b],[b,a]]</p><p><strong>Step 2:</strong> Calculate each element:</p><p>• (A²)₁₁ = a·a + b·b = a² + b²</p><p>• (A²)₁₂ = a·b + b·a = 2ab</p><p>• (A²)₂₁ = b·a + a·b = 2ab</p><p>• (A²)₂₂ = b·b + a·a = a² + b²</p><p><strong>Step 3:</strong> Match with given form [[α,β],[β,α]]:</p><p>A² = [[a² + b², 2ab],[2ab, a² + b²]]</p><p>∴ <strong>α = a² + b² and β = 2ab</strong></p>
Correct Answer: B