Pedagogy of Teaching the Inverse Matrix Method

I am teaching a group of (ordinary rather than honours) second-year engineers and we are studying matrices. I told the class today that as far as I could see we were only studying matrices and, particularly, the inverse matrix method as an introduction to more advanced matrix methods that would be studied in future.

However, the maths modules that they take in their next, final, third year are differential equations (no linear systems of differential equations) and, well, probability and statistics.

The only use that I can see that this group have for matrices is for solving linear systems. I know that there are plenty of more reasons to study matrices and in particular matrix inverses but this cohort will not see them.

It obviously strikes me as odd that the syllabus would recommend that we use the Inverse Matrix Method rather than the full Gaussian elimination theory.

Therefore my question is:

Assuming that we want to solve a linear system $A\mathbf{x}=\mathbf{b}$, what advantages, if any, does the inverse matrix method have over the full Gaussian elimination theory.

Thank you in advance for any answers; I am struggling to find one!

Solution 1:

When I teach linear algebra (it has been some years now), I always tell my students to never ever compute the inverse of a matrix, at least not if the matrix is much bigger than $3\times3$. If you need to solve $Ax=b$ for just one single $b$, do Gaussian elimination. If you need to do it for several $b$ values but a single $A$, compute the $LU$ decomposition of $A$ and use that to compute the required solutions.