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eigenvalues of a nxn matrix

Source: 26th annual VJIMC (2016), Category II, Problem 3

April 10, 2016
linear algebramatrixcollege contestseigenvalue

Problem Statement

For n≥3n \geq 3 find the eigenvalues (with their multiplicities) of the n×nn \times n matrix
[101000……00020100……00102010……00010201……00001020……00000102……00⋮⋮⋮⋮⋮⋮⋱⋮⋮⋮⋮⋮⋮⋮⋮⋱⋮⋮000000……20000000……01]\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0\\ 0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0\\ 1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0\\ 0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0\\ 0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & & \vdots & \vdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1 \end{bmatrix}
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