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The Identity Matrix and Inverses

Reference > Mathematics > Algebra > Matrices
 

In normal arithmetic, we refer to 1 as the "multiplicative identity." This is a fancy way of saying that when you multiply anything by 1, you get the same number back that you started with. In other words, 2 • 1 = 2, 10 • 1 = 10, etc.

Square matrices (matrices which have the same number of rows as columns) also have a multiplicative identity.

Would you like to see the 2 x 2 multiplicative identity matrix? It looks like this:

I2 =
1
0
0
1

Let's look at an example:
7
6
8
-2
1
0
0
1
=
7
6
8
-2

You see how the multiplicative identity gives right back to you the matrix you started with?

It turns out that the multiplicative matrices for 3 x 3, 4 x 4, etc. are all very similar; they have ones down the main diagonal, and zeroes everywhere else:

I3 =
1
0
0
0
1
0
0
0
1
; I4 =
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
.

So what is an inverse matrix? Technically, when we are talking about an inverse matrix, we are talking about a multiplicative inverse matrix.

When you're dealing with numbers, here are some multiplicative inverse examples: 5 is the multiplicative inverse of
1
5
and
2
3
is the multiplicative inverse of
3
2
. Why are they multiplicative inverses? Because when you multiply them together, you get the multiplicative identity (one).

The same is true of matrices:

If A is a 2 x 2 matrix, and A-1 is its inverse, then AA-1 = I2.

In arithmetic, there is one number which does not have a multiplicative inverse. That number is zero, because
1
0
is undefined. With matrices, there are many matrices which don't have inverses. There are two reasons why a matrix may not have an inverse:

  1. It is not square; only square matrices have inverses.
  2. Its determinant (check out the unit on Determinants for more information on evaluating the determinant of a matrix) is zero.

You might wonder what determinants have to do with inverses of matrices, and I can explain that in a loose way with an example. Suppose you wanted to find the inverse of the matrix
3
1
5
2
. One way to do it would be to set up an equation:

3
1
5
2
a
b
c
d
=
1
0
0
1

This matrix equation will give you a set of four equations in four unknowns:

3a + 1c = 1
3b + 1d = 0
5a + 2c = 0
5b + 2d = 1

A system of four equations with four unknowns...from our unit on determinants, you know that one of the ways to solve such a system is with Cramer's Rule, and the only time there is no solution is if the determinant has a zero value. Thus, when the determinant is zero, there is no set of 4 numbers that produces an inverse.

Fortunately, someone has gone to the trouble of creating a mini-formula/algorithm for you, to save you having to use Cramer's Rule every time you want to find the inverse of a 2 x 2 matrix.

If your matrix is
w
x
y
z
, and you want to find its inverse, you do the following:

Evaluate d =
w
x
y
z

Create a new matrix that looks like this:
z
d
-x
d
-y
d
w
d

This new matrix is the inverse of the original matrix. Notice that the w and z have switched places, and the x and y have become negative.

Are there methods for finding the inverses of 3 x 3 matrices? 4 x 4 matrices? Yes, there are. In fact, back in the dark ages of my high school days I wrote a three-page process proof for finding the inverse of any n x n matrix. It made me feel good, but it's not terribly practical in the days when computers can handle those horrifically complex calculations. For our purposes here, it is enough to show you (as we did above) how you would go about manually finding the inverse of a 2 x 2 using systems of equations, as well as the algorithmic short cut!

Questions

1.
Can 5 x 5 matrices have inverses?
2.
Can 4 x 3 matrices have inverses?
3.
Does the matrix
2
5
1
3
have an inverse? How do you know?
4.
Does the matrix
2
6
1
3
have an inverse? How do you know?
5.
What is the inverse of I2 (I2 =
1
0
0
1
)
6.
What are the entries in the inverse of
2
6
1
4
? List them clockwise starting with the first row, first column.
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