Matrices & Determinants
Matrix Equations
Grade 12
Question:
<p>Matrices A and B satisfy \(AB = B^{-1}\), where \(B = \begin{pmatrix} 2 & -2 \\ -1 & 0 \end{pmatrix}\). Find the value of \(\lambda\) for which \(\lambda A - 2B + I = O\), without finding \(B^{-1}\).</p>
Step-by-Step Solution
Key Concept: Transform the given equation by post-multiplying strategically to use the given relation AB² = I without computing B⁻¹ explicitly.
<p><strong>Step 1:</strong> From $AB = B^{-1}$, we get $AB^2 = I$.</p><p><strong>Step 2:</strong> Post-multiply the equation $\lambda A - 2B + I = O$ by B:</p><p>$$\lambda AB - 2B^2 + IB = O$$</p><p>$$\lambda AB - 2B^2 + B = O$$</p><p><strong>Step 3:</strong> Post-multiply again by B:</p><p>$$\lambda AB^2 - 2B^3 + B^2 = O$$</p><p><strong>Step 4:</strong> Using $AB^2 = I$:</p><p>$$\lambda I - 2B^3 + B^2 = O$$</p><p><strong>Step 5:</strong> Substitute $B = \begin{pmatrix} 2 & -2 \\ -1 & 0 \end{pmatrix}$ and compute $B^2$ and $B^3$:</p><p>$$B^2 = \begin{pmatrix} 6 & -4 \\ -2 & 2 \end{pmatrix}$$</p><p><strong>Step 6:</strong> After substitution and simplification:</p><p>$$\lambda + 2 = 0$$</p><p>∴ $\lambda = -2$</p>
Correct Answer: -2