I wrote a text book (Springer-Verlag GTM 216). Therefore I am potentially concerned by your statement I was dissatisfied with the textbook I followed.

Because my book is a bit advanced, I do not advise you to follow it. But let me extract a few topics that can be taught in the context you describe.

• The spectral radius. Matrix norms.
• The numerical radius, Toepliz-Hausdorff theorem.
• Non-negative matrices, Perron-Frobenius theorem.
• Matrix exponential, its use for ODEs.
• Positive definite Hermitian matrices, their square root, the polar decomposition.
• The Schur complement and Sherman-Morison formulae.
• Elementary methods for solving linear systems.