Matrices & Determinants
Matrix multiplication and powers
Grade 12
Question:
<p>If <span>\(A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}\)</span> and <span>\(A^2 = \begin{bmatrix} \alpha & \beta \\ \beta & \alpha \end{bmatrix}\)</span> then</p>
<p>(a) \(\alpha = a^2 + b^2,\ \beta = ab\)</p>
<p>(b) \(\beta = a^2 + b^2,\ \alpha = 2ab\)</p>
<p>(c) \(\alpha = a^2 + b^2,\ \beta = 2ab\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: When squaring a symmetric matrix of the form [a b; b a], the result maintains the same symmetric structure. Compute A² directly using matrix multiplication to find relationships between (a,b) and (α,β).
<p><strong>Step 1:</strong> Compute A² using matrix multiplication:</p><p>A² = [a b; b a] × [a b; b a]</p><p><strong>Step 2:</strong> Calculate entries:</p><p>Entry (1,1): a·a + b·b = a² + b² = α</p><p>Entry (1,2): a·b + b·a = 2ab = β</p><p>Entry (2,1): b·a + a·b = 2ab = β</p><p>Entry (2,2): b·b + a·a = a² + b² = α</p><p><strong>Step 3:</strong> Verify symmetric form is preserved and identify:</p><p>α = a² + b²</p><p>β = 2ab</p><p>∴ Answer: C (relationships are α = a² + b² and β = 2ab)</p>
Correct Answer: C