<p>If \(A = \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix}\), \(B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}\) and \((A+B)^2 = A^2 + B^2 + 2AB\), then</p>
Step-by-Step Solution
Key Concept: The condition (A+B)² = A² + B² + 2AB holds if and only if AB = BA (matrices commute). Use this commutativity requirement to set up equations for a and b.
<p><strong>Step 1:</strong> Expand (A+B)²</p><p>(A+B)² = A² + AB + BA + B²</p><p>Given condition: (A+B)² = A² + B² + 2AB</p><p>Therefore: A² + AB + BA + B² = A² + B² + 2AB</p><p><strong>Step 2:</strong> Simplify to get commutativity condition</p><p>AB + BA = 2AB</p><p>BA = AB</p><p>Matrices A and B must commute.</p><p><strong>Step 3:</strong> Calculate AB</p><p>AB = \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix} = \begin{bmatrix} a-b & 1+1 \\ 2a+b & 2-1 \end{bmatrix} = \begin{bmatrix} a-b & 2 \\ 2a+b & 1 \end{bmatrix}</p><p><strong>Step 4:</strong> Calculate BA</p><p>BA = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} a+2 & -a+1 \\ b-2 & -b-1 \end{bmatrix}</p><p><strong>Step 5:</strong> Set AB = BA and solve</p><p>From (1,1) entry: a - b = a + 2 → b = -2</p><p>From (1,2) entry: 2 = -a + 1 → a = -1</p><p>From (2,1) entry: 2a + b = b - 2 → 2a = -2 → a = -1 ✓</p><p>From (2,2) entry: 1 = -b - 1 → b = -2 ✓</p><p><strong>Step 6:</strong> Verify consistency</p><p>a = -1, b = -2 satisfies all four equations simultaneously.</p><p>∴ Answer: A (a = -1), D (b = -2)</p>
Correct Answer: A,D