I am attempting to find the minimal polynomial of a matrix. My characteristic equation turns out to be $x^3 - x$ which factors out to $x(x-1)(x+1)$. Now, I am reading that the minimal polynomial is defined as
The unique polynomial of smallest degree which when evaluated at the matrix A is the zero matrix, and I also read that is it a factor of the characteristic equation. So, it must be $x, x-1,$ or $x+1$. But, I do not understand how to compute $A-1$ nor $A + 1$, because we are adding an integer to a matrix.
You must replace all numbers in the equation with multiples of the identity matrix. So A+1 is really A+I, and A+2 is A+2I.