linear algebra - Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors ...

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them?

So if I calculated determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors, what are ways to be sure that I didn't do a major mistake? I don't want to verify my solutions all the way through, I just want a quick way which gives me that it is highly likely that the calculated determinant is right etc.

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Let $A$ be a matrix $A \in \operatorname{Mat}(n, \mathbb{C})$,

let $\det(A)$ be the determinant of matrix $A$,

let $v_1, v_2, ..., v_k$ be eigenvectors of matrix $A$,

let $\lambda_1, \lambda_2, ..., \lambda_n$ be eigenvalues of matrix $A$,

let $\chi_A(t) = t^n + a_{n-1}t^{n-1}+\cdots + a_0 = (t-\lambda_1)\cdots(t-\lambda_n)$ be the characteristic polynomial of matrix $A$,

let $\mu_A(t)$ be the minimal polynomial of matrix $A$.

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Verifications suggested so far:

eigenvectors / eigenvalues

• $\det(A) = \lambda_1^{m_1} \lambda_2^{m_2} \cdots \lambda_n^{m_l}$ where $m_i$ is the multiplicity of the corresponding eigenvalue


• $a_0 = (-1)^n\lambda_1\cdots\lambda_n$


• eigenvectors can be verified by multiplying with the matrix; the eigenvalues can be verified at the same time; i.e. $A v_i = \lambda_i v_i$

determinant

• $\det(A) = \lambda_1^{m_1} \lambda_2^{m_2} \cdots \lambda_l^{m_l}$ where $m_i$ is the multiplicity of the corresponding eigenvalue

characteristic / minimal polynomial

• $a_0 = (-1)^n\lambda_1\cdots\lambda_n$


• $\mu_A(A) = 0$ and $\chi_A(A) = 0$


• $\mu_A(t) \mid \chi_A(t)$