Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 2 / 6G (Linear Mathematics)

The minimal polynomial divides the annihilating polynomial $t^2(t-1)$, since $A^3-A^2=0$. Every nonconstant monic divisor can occur in dimension four: use Jordan blocks of sizes at most two at eigenvalue $0$ and of size one at eigenvalue $1$, adding scalar blocks as needed. The complete list isFor example the first four occur for $0$, $J_2(0)\oplus 0\oplus 0$, $I$, and $\mathrm{diag}(0,0,1,1)$, respectively; the final one occurs for $J_2(0)\oplus 0\oplus 1$.

Failure to be diagonalizable requires a size-two Jordan block at zero. The condition $A^2\ne 0$ requires at least one block at one, since every allowed zero block squares to zero. A size-two zero block and one scalar block at one leave only one dimension. It can carry zero or one. Thus, up to permutation of blocks, the only possible Jordan normal forms are