Let A = [2 1; 1 2]. Compute the top left entry of A⁵, that is, the entry in row 1, column 1.

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Power of a symmetric matrix

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Let $A = [2112]$ . Compute the top left entry of $A^{5}$ , that is, the entry in row $1$ , column $1$ .

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Hint

Split the vector $(1, 0)$ along the eigenvectors of $A$ , which are easy to guess.

Solution

Since $A (1, 1) = 3 (1, 1)$ and $A (1, - 1) = (1, - 1)$ , the eigenvalues are $3$ and $1$ .
Write $(1, 0) = \frac{1}{2} \times (1, 1) + \frac{1}{2} \times (1, - 1)$ ; then $A^{5} (1, 0) = \frac{1}{2} \times 3^{5} \times (1, 1) + \frac{1}{2} \times (1, - 1)$ .
The first coordinate of $A^{5} (1, 0)$ is the entry in row $1$ , column $1$ of $A^{5}$ .
It equals $\frac{243 + 1}{2} = 122$ .

Answer: $122$