<p>Given \(P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}\) and \(Q - P^5 = I_3\). Find \(Q\).</p>
Step-by-Step Solution
Key Concept: Recognize P as a lower triangular matrix with 1's on the diagonal (unipotent matrix). Use the binomial theorem property: for such matrices, P^n can be computed by observing the nilpotent structure of (P - I), then solve Q = P^5 + I.
<p><strong>Step 1:</strong> Write P = I + N where N = P - I = <span style='font-family:monospace'>[[0,0,0],[3,0,0],[9,3,0]]</span> is strictly lower triangular (nilpotent).</p><p><strong>Step 2:</strong> Since N is strictly lower triangular of size 3×3, N³ = 0. By binomial expansion: P⁵ = (I + N)⁵ = I + 5N + 10N² + 10N³ + ... = I + 5N + 10N²</p><p><strong>Step 3:</strong> Compute N²:<br/>N² = <span style='font-family:monospace'>[[0,0,0],[0,0,0],[27,9,0]]</span></p><p><strong>Step 4:</strong> Calculate P⁵ = I + 5N + 10N²:<br/>P⁵ = <span style='font-family:monospace'>[[1,0,0],[3,1,0],[9,3,1]]</span> + <span style='font-family:monospace'>[[0,0,0],[15,0,0],[45,15,0]]</span> + <span style='font-family:monospace'>[[0,0,0],[0,0,0],[270,90,0]]</span><br/>= <span style='font-family:monospace'>[[1,0,0],[18,1,0],[324,108,1]]</span></p><p><strong>Step 5:</strong> From Q - P⁵ = I₃, we get Q = P⁵ + I₃ = <span style='font-family:monospace'>[[2,0,0],[18,2,0],[324,108,2]]</span></p><p>∴ Answer: A</p>
Correct Answer: A