Example 2

  MatrixForm[B]

The following finds the characteristic polynomial of B.

  Det[B-x IdentityMatrix[4]]

  Factor[Det[B-x IdentityMatrix[4]]]

Some shortcuts.

  Bm2I=B-2IdentityMatrix[4];
  Bm5I=B-5IdentityMatrix[4];

Let's check (-5+x)(-2+x).

  MatrixForm[Bm5I.Bm2I]

Let's check (-5+x)(-2+x)^2.

  MatrixForm[Bm5I.Bm2I.Bm2I]

By default (and the Cayley-Hamilton Theorem), the minimal polynomial is the characteristic polynomial (suitably written to be monic):
(x-2)^3 (x-5)

Recall the Jordan form of B.

  MatrixForm[JB]

Compare the size of the Jordan block for each eigenvalue with the multiplicity of that eigenvalue in the MINIMAL polynomial.

Up to Jordan Form and Minimal Polynomials