### Quiz: Describe How and Why Math Works

Math test today. I'm tired. We were quizzed on solving systems of linear equations. A lot of the test focused on using matrices. We were allowed to use calculators for part of the test.

I think I understand pretty well how to add, subtract, and multiply matrices as well as how to divide matrices by multiplying a matrix by its inverse. (We'll see how well I understand when I get the test back!)

But I have been a little confused about why some of the math works. In particular, I'm not entirely solid on the process for finding inverse matrices. Without getting too much in the weeds, to figure out the inverse of a matrix you have to find something called a determinant and then plug it into a formula that involves changing the position or the negative signs within the matrix. Forgive the oversimplification. (Or feel free to put it in words better than I just did!)

It's difficult to describe how or why math works. It's easier to just write the formula and say, "Do this." Several readers have commented on this blog that what's often missing from math education is more of a focus on why certain applications work. I agree. It's harder to remember what to do, if you don't have some sense of why it works.

A worksheet that the teacher, Tricia Colclaser, handed out after the test came close to addressing this. Titled "Math Journal," it asked us to explain how to determine whether two matrices can be added or subtracted or multiplied. Colclaser asked the class to "explain in a sentence, a full complete sentence."

Here's a primer on inverse matrices and a look at the formula for finding them - if you want to review.

By Michael Alison Chandler  |  October 16, 2008; 10:51 AM ET  | Category:  Class Time
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I think part of the trick with understanding inverse matrices is to relate it back to the algebra you already know. You know how to solve 3*x = 12. However, although you're first answer is probably, "divide both sides by three", another way to look at it is "multiply both sides by the INVERSE of three". The trick is to remember what an inverse is: when you multiply an item by it's inverse, you get 1, the multiplicative identity. For matrices, the multiplicative identity is the Identity matrix (1's on the diagonal and zero's elsewhere). The inverse of a matrix is another matrix, so that when you multiply them together, you get the Identity matrix. This lets you solve systems like A*x = b, where the matrix A and vector b are given, in the same manner you solved 3*x = 12. There is no division - just multiplication.

Posted by: notea42 | October 16, 2008 12:51 PM | Report abuse

Additionally, mucking around with determinants and sign-flipping seems like a strange way to invert matrices. I've always found Gaussian-Elimination much easier to teach to people - see and

Posted by: notea42 | October 16, 2008 12:54 PM | Report abuse

I remember getting on my high school teachers nerves as I always wanted to know the "why" behind what we were doing. One day she had enough and simply exclaimed "Because that is how it is done, that's why!" She later explained in private that my constant asking of why created two problems for her: 1) she often had to rack her brain to remember the answer, which made her self-conscious in front of a room of students, and 2) the "why" was often a result of higher mathematics, which makes it very difficult to explain without a knowledge of more advanced mathematics. This did make my life easier, because rather than trying to figure out why to everything I just accepted that was how to do it.

Another example of this is when I was in college calculus and the professor explained Pi using calculus, and from my memory, something that can't be done using lower forms of math.

Posted by: mskidz | October 16, 2008 4:50 PM | Report abuse

The formula that you are asking about comes directly from the definition of the inverse. The matrix X is the inverse of the matrix A if it satisfies the equation AX=I, where I is the identity matrix. For the 2x2 matrix case, you can solve for this set of 4 equations for the elements of X in terms of the elements of A, which gives the formula that you describe.

Posted by: quinblue1 | October 16, 2008 8:41 PM | Report abuse

I'm not sure what was meant by the comment that "the professor explained Pi using calculus, and from my memory, something that can't be done using lower forms of math."

Pi is very simple; it is simply what you get when you divide any circle's circumference by its diameter. That's it - no calculus necessary.

Posted by: BradJolly | October 17, 2008 12:08 AM | Report abuse

I was also taught the Gaussian elimination method in 8th or 9th grade by a college professor (math team coach). It seems a LOT easier to solve these problems when you can actually muck around with the actual equations than simply plugging numbers into a memorized formula.

I happen to think many public school teachers simply do not have the background to understand what they are teaching. It's a pity because I think many people would enjoy math much more if taught by sufficiently educated and well-trained teachers.

Posted by: slackermom | October 17, 2008 12:20 PM | Report abuse

I'm with Michael on this one. In college I puzzled over the reason why the general determinant forumla works for matrix inversion -- for the 2x2 matrix it is easy to see. Not so much for the general case. I still have that feeling after getting a doctorate in statistics. It was one of the few times I didn't have much of an intuitive feeling for why something worked.

The emphasis probably should be on Gaussian elimination as suggested by slackermom.

Posted by: fedbert | October 17, 2008 12:53 PM | Report abuse

It seems to me that inversion, and the determinant, is an example of one of those things that's hard to explain without higher mathematics.

If we view the matrix as a kind of spatial transformation, then the determinant is a "scale factor" that blows up the space. Thus, if you're reversing the process, you have to "shrink" by the same amount, and hence you need to divide by the determinant. All of this can be made more precise, but it would completely overwhelm high school students I think.

Posted by: geomblog | October 17, 2008 11:07 PM | Report abuse

In mathematics, some things are true for succinct and elegant reasons and others are true simply because the calculations turn out that way. In this particular case of the inverse of the 2 by 2 matrix, I believe it to be the latter. As others have hinted, there are things to be understood about the inverse which can make one feel more comfortable with the overall form but it requires a lot of higher math, all of which any beginner is sure to find disconcerting. In this type of situation, the calculation turns out to be very close to the best explanation.

Multiplying the vector (x,y) by the matrix A gives us (ax+by,cx+dy). Ideally we want the inverse of A to do the opposite. It should produce the numbers (x,y) even if we do not know them in advance.

When we get the vectors that we have to invert, we don't know what x and y are. We get two numbers like (w,z) which each contain a contribution derived from our original x and y. What we know is that the first coordinate has a portion which is b times the original y and the second coordinate contains a contribution that is d times the original y. A clever idea might be to multiply the first coordinate by d and the second coordinate by b. In this way, We produce two numbers for which the contributions due to y are the same. Now, one more bright idea is to subtract like this:

Following similar logic we soon get (ad-bc)*y. In order to get our orginal numbers, we notice that we must divide by ad-bc. If we take care to write all this down as a matrix, we get the inverse of A. Multiplying by the inverse of A does the opposite of what A does, which makes sense.

If the ideas seem a bit strange or awkward, give yourself some time to acclimate.

"In mathematics you don't understand things. You just get used to them."
- John von Neuman

Posted by: mathlete | October 20, 2008 1:20 AM | Report abuse

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