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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!

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A great advantage of the inverse matrix method is that, when you have to solve $Ax=b$ and then $Ax=b'$, you do the job only once. – 1015 Feb 12 '13 at 22:08
The inverse matrix method of solution mimics what they know from the scalar case: to solve $ax=b$, multiply both sides by $a^{-1}$. (This potentially works both ways: it can help with understanding the inverse matrix method, and can help with understanding division as multiplication by a multiplicative inverse.) – user12477 Feb 12 '13 at 22:11
Yeah, basically with the inverse matrix method, you end up with both the solution and the inverse matrix ! – Damien L Feb 12 '13 at 22:11
@user12477 yes I did appreciate this point at the time... when I asked them how to solve the matrix equation $AX=B$ from what they knew about the equation $ax=b$ someone said "bring over..." or words to that effect. I was able to talk to them about algebra at this point. – Jp McCarthy Feb 12 '13 at 22:16
The whole art of numerical linear algebra is to solve your problem without inverting a matrix, or multiplying two matrices together. – Chris Godsil Feb 12 '13 at 22:40
up vote 2 down vote accepted

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.

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Thank you Harold. It looks like I will move to change the module content. This kind of thing is daft; it doesn't help the students nor the one teaching the material. – Jp McCarthy Feb 12 '13 at 23:06

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