The product of the non-zero eigenvalues of the matrix $$\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$$ is _________.

Let the given $5 \times 5$ matrix be $A$.

Notice the structure of the rows:

• Row 1 and Row 5 are identical: $[1, 0, 0, 0, 1]$

• Row 2, Row 3, and Row 4 are identical: $[0, 1, 1, 1, 0]$

Since there are only $2$ linearly independent rows, the rank of $A$ is $2$. Hence, exactly $2$ eigenvalues are non-zero, and the remaining $3$ eigenvalues are $0$.

We can find the non-zero eigenvalues by testing eigenvectors spanned by the independent row patterns:

1. Let $v_1 = [1, 0, 0, 0, 1]^T$:

2. Let $v_2 = [0, 1, 1, 1, 0]^T$:

Thus, the non-zero eigenvalues of the matrix are $2$ and $3$.

Their product is: