What is the minimal Polynomial of matrix A of order $3\times3$ where $a_{11}=1$ and all other elements are zeroes? I know that the minimal polynomial shouldn't be expressed using a matrix, so how can I make the element $a11$ equal to zero?

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 $A^2-A=0$. 
