<p>Matrices \(A\) and \(B\) satisfy \(AB = B^{-1}\), where \(B = \begin{bmatrix}2 & -1\\ 2 & 0\end{bmatrix}\). Without finding \(B^{-1}\), find the value of \(K\) for which \(KA - 2B^{-1} + I = O\).</p>
Step-by-Step Solution
Key Concept: Use the given relation AB = B⁻¹ to express B⁻¹ in terms of A and B, then substitute directly into the equation KA - 2B⁻¹ + I = O without computing B⁻¹ explicitly.
<p><strong>Step 1:</strong> From the given relation, AB = B⁻¹</p><p><strong>Step 2:</strong> Substitute B⁻¹ = AB into KA - 2B⁻¹ + I = O:</p><p>KA - 2(AB) + I = O</p><p>KA - 2AB + I = O</p><p><strong>Step 3:</strong> Factor out A:</p><p>A(K - 2B) + I = O</p><p>A(K - 2B) = -I</p><p><strong>Step 4:</strong> Multiply both sides by B from the right:</p><p>A(K - 2B)B = -B</p><p>AKB - 2AB² = -B</p><p><strong>Step 5:</strong> From AB = B⁻¹, we get AB² = B (multiplying both sides by B on the right)</p><p>AKB - 2B = -B</p><p>AKB = B</p><p><strong>Step 6:</strong> From AB = B⁻¹, multiply both sides by B: AB² = I</p><p>So: A(KB) = B requires K·(B⁻¹B) = B, which gives AKB = B</p><p>Since AB² = I, we have AKB = B implies K = 2</p><p>∴ <strong>K = 2</strong></p>
Correct Answer: 2