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My government has told me that I must teach 17 year olds how to invert a 3 by 3 matrix using adjoints/cofactors, without using any technology.

  1. Is there any reason why you would want to know how to do this by hand? (As opposed to on calculator)

  2. When we went through the step-by-step recipe ("And now, you change the signs of these entries...") the students wanted to know why it works. I am loath to hand over a magic recipe without justifying, but am unsure of what to say. What could I say?

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    $\begingroup$ See this video series (video 8 is about inverses) for some inspitration on explaining why the calculations are the way they are. As for why they should learn it? They shouldn't. Almost no concrete piece of math they learn at that stage is something they should learn (unless they want a career in something mathematical). That being said, learning new mathematics is important for their logical sense, and linear algebra is as good a field as any. $\endgroup$
    – Arthur
    Oct 3, 2017 at 9:01
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    $\begingroup$ Why don't you ask your government instead? IMHO using adjoints/cofactors is a bad idea. The fact that you can't tell them why it works is one clue that it's a bad idea. Another clue is that it's more complex than Gauss elimination in most cases. $\endgroup$
    – skyking
    Oct 3, 2017 at 9:02
  • $\begingroup$ Also, a focus on calculation algorithms in education is a bad thing in my opinion. They need to know some step-by-step recipes, but too many of them and too much time spent practicing them is both boring and not what mathematics is about. $\endgroup$
    – Arthur
    Oct 3, 2017 at 9:03
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    $\begingroup$ Vote to replace your government. $\endgroup$
    – uniquesolution
    Oct 3, 2017 at 9:23
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    $\begingroup$ If the topic inspires students asking “why,” then the topic is worthy of being presented. $\endgroup$
    – user52817
    Oct 3, 2017 at 17:43

2 Answers 2

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In computer graphics, the view matrix is the inverse of the camera matrix. This is needed, e.g., in game programming.

In general, matrices are used to convert from coordinate system A to coordinate system B, and the inverse converts in the opposite direction, from B to A. There are circumstances where both are needed.

          ProjMat
          (Image from UC Santa Barbara, CS180 Intro Comp Graphics.)
Generally the matrices used in computer graphics are $4 \times 4$—homogenous coordinates. But that is a minor difference.

So this motivates the need for the inverse ("Why bother calculating the inverse ..."?). You could not program this without knowing how to do it "by hand" (1). So a motivation for understanding it at that level is that detailed knowledge is needed to actually implement the inverse. A motivation for understanding the concept is that one cannot become a game programmer without understanding matrix inverses.

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It shows why every matrix with non-zero determinant has an inverse, and vice-versa, every invertible matrix has non-zero determinant. So there is a quick calculation to decide whether three simultaneous equations have a unique solution.

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