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

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 first few powers of \( P \): \[ P^2 = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix}. \] \[ P^3 = \begin{pmatrix} 5 & 4 \\ 4 & 5 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 14 & 13 \\ 13 & 14 \end{pmatrix}. \] \[ P^4 = \begin{pmatrix} 14 & 13 \\ 13 & 14 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 39 & 41 \\ 41 & 39 \end{pmatrix}. \] \[ P^5 = \begin{pmatrix} 39 & 41 \\ 41 & 39 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 119 & 119 \\ 119 & 119 \end{pmatrix}. \] The trace of \( P^5 \) is the sum of the diagonal elements: \[ \text{trace}(P^5) = 119 + 119 = 238. \] Step 3: Calculating \( \text{trace}(Q^4) \)
Next, we calculate the powers of \( Q \): \[ Q^2 = \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -1 & 5 \\ -6 & 15 \end{pmatrix}. \] \[ Q^3 = \begin{pmatrix} -1 & 5 \\ -6 & 15 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -11 & 29 \\ -24 & 60 \end{pmatrix}. \] \[ Q^4 = \begin{pmatrix} -11 & 29 \\ -24 & 60 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -59 & 147 \\ -120 & 300 \end{pmatrix}. \] The trace of \( Q^4 \) is the sum of the diagonal elements: \[ \text{trace}(Q^4) = -59 + 300 = 241. \] Step 4: Final Calculation
Now we can add the traces of \( P^5 \) and \( Q^4 \): \[ \text{trace}(P^5 + Q^4) = 238 + 241 = 341. \] Thus, the value of \( \text{trace}(P^5 + Q^4) \) is \( \boxed{341} \).