Originally Posted by
HallsofIvy Can't you use the fact that
1) addition of matrices is commutative.
2) addition of matrices is associative.
3) there exist a "0" matrix, the additive identity.
4) every matrix has an additive inverse.
5) multiplication of matrices is associative.
6) there exist a "1" matrix, the multiplicative identity.
are true of ALL matrices? so the only thing left to show is that the every non-zero matrix is invertible.
(Well, I guess you should show that the 0 matrix, $\displaystyle \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$ is of the given form, with a= b= 0 and that the identity matrix, $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, is of that form with a= 1, b= 0.)