For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...
Let $R$ be a ring. Then I would like to know the units in $M_n(R)$. Here is my idea. The determinant is a homomorphism from $M_n(R)$ to $R$. If $A$ is a unit, then there is a $B$ such that $AB = I$. ...