What is the minimal Polynomial of matrix A of order $3\times3$ where $a_{11}=1$ and all other elements are zeroes?

Question

I know that the minimal polynomial shouldn't be expressed using a matrix, so how can I make the element $a11$ equal to zero?

Answer 1

The minimal polynomial is $p(t)=t^2-t$. This is the polynomial of least degree with roots $0$ and $1$ (which are the eigenvalues of $A$). That in itself is not enough to make it the minimal polynomial, but one can easily check that $A^2-A=0$.

Answer 2

You can find the minimal polynomial quite easily by taking powers of $A$. $A^2$ is easily calculated to be equal to $A$, so $A^2=A$ and thus $A^2-A=0$. Since this factors into $A(A-I)=0$, and $A$ satisfies neither $A=0$ nor $A-I=0$, the polynomial $t^2-t$ is minimal. Note that since o is an eigenvalue of multiplicity two, the characteristic polynomial is $t^3-t^2$ (assuming, by "order 3*3", you mean a 3x3 matrix).