TITLE: How do you compute eigenvalues/vectors of big $n\times n$ matrix?
QUESTION [7 upvotes]: The product of the non-zero eigenvalues of the matrix is ____.
$$\pmatrix{1&0&0&0&1\\0&1&1&1&0\\0&1&1&1&0\\0&1&1&1&0\\1&0&0&0&1}$$

My attempt:
Well, answer is $6$. It's a big matrix (but not more), so finding eigenvalues by characteristics equation will be lengthy process. I'm trying for any short trick to find eigenvalues of a big $n \times n$ matrix.
This explained "eigenvalues by inspection". I'm not getting properly.

Can you explain, eigenvalues and eigenvectors by inspection for this matrix in steps please?

Can you explain in steps.

REPLY [2 votes]: You can find the characteristic polynomial, and then the last nonzero coefficient is $\pm$ the product of the non-zero eigenvalues, depending on the degree. 
For example, the characteristic polynomial for above is:
$$p(\lambda)=\det(\lambda I-A) = \lambda^5-5\lambda^4+6\lambda^3,$$ the product of the non-zero eigenvalues would be $6$. 
This is only true, in general, if you want to count with duplication, so if there is a repeated non-zero eigenvalue, then you include those in the product multiple times.
You can get the non-repeated product of the non-zero eigenvalues by computing the GCD of the two polynomails, $p(x),p'(x)$, and dividing that from $p(x)$. So:
$$q(x)=\frac{p(x)}{\gcd(p(x),p'(x))}$$
is a polynomial with the same roots as $p(x)$, but no repeated roots. Then the product of the non-zero roots, without multiplicity, will be $\pm$ the last non-zero coefficient of $q(x)$.