The Identity Matrix and InversesReference > 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 = Let's look at an example: =
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 = ; I4 = .
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
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
- It is not square; only square matrices have inverses.
- Its determinant (check out the unit on Determinants for more information on evaluating the determinant of a matrix) is zero.
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 , and you want to find its inverse, you do the following: Evaluate d = Create a new matrix that looks like this:
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!