Question

Let \( P = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \) and \( Q = \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} \). Then the value of \( \text{trace}(P^5 + Q^4) \) equals:

Hint:

To compute the trace of matrix powers, first calculate the matrix powers and then sum the diagonal elements. The trace function is linear, so you can separate the trace of the sum of matrices.

Answer 1

Correct Answer: 341

Step 1: Trace of a Matrix
Recall that the trace of a matrix is the sum of its diagonal elements. The trace function is linear, so: \[ \text{trace}(P^5 + Q^4) = \text{trace}(P^5) + \text{trace}(Q^4). \] Step 2: Calculating \( \text{trace}(P^5) \)
First, we compute the successive powers of \( P \): \[ P^2 = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix}. \] \[ P^3 = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 14 & 13 \\ 13 & 14 \end{pmatrix}. \] \[ P^4 = \begin{pmatrix} 14 & 13 \\ 13 & 14 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 39 & 41 \\ 41 & 39 \end{pmatrix}. \] \[ P^5 = \begin{pmatrix} 39 & 41 \\ 41 & 39 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 119 & 119 \\ 119 & 119 \end{pmatrix}. \] Thus, \[ \text{trace}(P^5) = 119 + 119 = 238. \] Step 3: Calculating \( \text{trace}(Q^4) \)
Now compute the powers of \( Q \): \[ Q^2 = \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -1 & 5 \\ -6 & 15 \end{pmatrix}. \] \[ Q^3 = \begin{pmatrix} -1 & 5 \\ -6 & 15 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -11 & 29 \\ -24 & 60 \end{pmatrix}. \] \[ Q^4 = \begin{pmatrix} -11 & 29 \\ -24 & 60 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -59 & 147 \\ -120 & 300 \end{pmatrix}. \] Hence, \[ \text{trace}(Q^4) = -59 + 300 = 241. \] Step 4: Final Calculation
\[ \text{trace}(P^5 + Q^4) = 238 + 241 = 479. \] Final Answer:
\[ \boxed{479} \]