 # I need a source of practice problems for finding inverse of matrix using row operations

I’m tutoring a kid in Algebra II. He’s having trouble with matrices. He is unable to report what exactly it is he needs to know about them for the test he’ll be (re)taking. So all I have to go on is his textbook.

The textbook has a chapter section or two on solving systems using matrices which seem to presuppose that the student knows a way to find inverses of 3x3 matrices. But going back to the chapter section on 3x3 inverses, I find only a method for verifying that two matrices are inverses, and no method given for actually finding the inverse. No surprise, since finding the inverse of a 3x3 matrix can be very difficult.

Yet as I said, future chapter sections seem to presuppose that the student should be able to find the inverse.

So one fairly easy method for finding inverses for certain tailor-made examples is the method using [url=http://www.purplemath.com/modules/mtrxinvr.htm]row operations*. When I briefly described this method to the student he said it “sounded familiar” which I take to be a clue that this is the method his teacher uses in class and will be using to generate inverse problems on the exam. So I’m thinking I’ll review the method with the student next time, but for that, I need some of those tailor-made examples I mentioned. Which brings me to the question–is there a good source online for such examples? I can’t find one.

I’m kind of reminding myself about matrices as we go on this one, so if something I’ve said above makes you think there’s some important info I’ve forgotten since high school that would be good. The textbook the student is using seems to be written with the expectation that a lot of explanation will come from the instructor from the course. As such, the book seems very incomplete in many ways. In any case, I was unable to glean much info from it–it seemed opaque, not because of density but because of missing presupposed knowledge.

Not online, unfortunately, but most introductory linear algebra textbooks should have some practice problems of this type. The text I have, Anton’s Elementary Linear Algebra, has about a dozen such practice problems. Here are the 3x3 matrices among them (the rest are 2x2 or 4x4):

``````

3  4 -1     -1  3 -4     1  0  1     2  6  6     1  0  1
1  0  3      2  4  1     0  1  1     2  7  6    -1  1  1
2  5 -4     -4  2 -9     1  1  0     2  7  7     0  1  0

1/5  1/5 -2/5       sqrt(2)  3 sqrt(2)  0
1/5  1/5  1/10    -4 sqrt(2)  sqrt(2)   0
1/5 -4/5  1/10         0        0       1

``````

Note that part of Anton’s pedagogy here is to teach the student that if you get a row of zeros on the left-hand side, the matrix is not invertible. So it may be that some of the matrices don’t have an inverse; you might want to check them ahead of time.

Shaum’s Outline Linear Algebra 3rd Edition provides the following matrices for practice:

``````

1   2  -4
-1 -1  5
2   7  -3
1  3  -4
1  5  -1
3  13 -6
(find inverse if it exists of the following):
1 -2 -1
2 -3 1
3 -4 4
1 2 3
2 6  1
3 10 -1

1 3 -2
2 8 -3
1 7 1

2 1 -1
5 2 -3
0 2 1

``````

Is it possible they’re expecting the students to use graphing calculators to find inverses? (That is, enter the matrix into the calculator and ask it for the inverse?)

I didn’t think to ask, but the student didn’t seem to think he should be using a calculator for any of this. I’ll ask him explicitly next time.

[/quote]
So one fairly easy method for finding inverses for certain tailor-made examples is the method using row operations.
[/quote]

You may already know this, but the examples need not be tailor made. This method works for any matrix–it finds the inverse if it exists; it shows there is no inverse when it does not exist.

Of course, if it’s not a “tailor-made” example to make the numbers work out nicely, the arithmetic can get kinda ugly, but the method still works.

As Cabbage says, this method works for any matrix. The numerically “easy” examples tend to be those for which you don’t end up with lots of fractions. One easy way to generate arbitrarily many of these examples is to just generate lots of matrices in O(3,Z), which is the group of matrices with integer entries and determinant +/- 1. For example, define some very simple matrices

``````

[ 1 1 0 ]       [ 1 0 1 ]       [ 1 0 0 ]       [ 0 1 0 ]       [ 1     ]
A = [ 0 1 0 ]   B = [ 0 1 0 ]   C = [ 0 1 1 ]   D = [ 1 0 0 ]   E = [   1   ]
[ 0 0 1 ]       [ 0 0 1 ]       [ 0 0 1 ]       [ 0 0 1 ]       [    -1 ]

``````

Multiplying these, their transposes, or their inverses together in any order gives an element of O(3,Z); since each of these matrices is trivial to invert (and also has only integer entries), if you know the order you multiplied them together you can generate their inverses just as easily, for an answer key.

Of course, you should also have some exercises which involve fractions after he’s mastered the basic idea; you could add another matrix F=diag([1,1,3]) to the mix, for example.

What, only for 3 x 3 matrices and not more generally? You could always just memorize a formula for it (though it’s not at all enlightening to do so).

So one fairly easy method for finding inverses for certain tailor-made examples is the method using row operations.
[/quote]

You may already know this, but the examples need not be tailor made. This method works for any matrix–it finds the inverse if it exists; it shows there is no inverse when it does not exist.

Of course, if it’s not a “tailor-made” example to make the numbers work out nicely, the arithmetic can get kinda ugly, but the method still works.
[/QUOTE]

Yeah, I meant in the sense of examples where the student doesn’t have to get bogged down in complicated fraction arithmatic. I wanted to start with some simple examples first.

However, I had forgotten the systematic way to do the row-operations method. I was instead just trying to sort of guess at the right operations to do. Since then I’ve reconstructed/remembered the systematic way to do it–divide the whole row by the number that puts a 1 in the right place, then add multiples of the row to the other rows to get zeros in the same column, rince repeat. Creaky math muscles are coming back into shape as we speak…