I am so glad you asked that.
One of the major uses of inverses is to solve a system of linear equations. You can write a system in matrix form as AX = B.
Real Numbers
When working in the real numbers, the equation ax=b could be solved for x by dividing both sides of the equation by a to get x=b/a, as long as a wasn't zero. It would therefore seem logical that when working with matrices, one could take the matrix equation AX=B and divide both sides by A to get X=B/A.
However, that won't work because ...
There is NO matrix division!
Ok, you say. Subtraction was defined in terms of addition and division was defined in terms of multiplication. So, instead of dividing, I'll just multiply by the inverse. This is the way that it has to be done.
So, what is the inverse of a matrix?
Well, in real numbers, the inverse of any real number a was the number a-1, such that a times a-1 equaled 1. We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn't zero.
The inverse of a square matrix A, denoted by A-1, is the matrix so that the product of A and A-1 is the Identity matrix. The identity matrix that results will be the same size as the matrix A. Wow, there's a lot of similarities there between real numbers and matrices. That's good, right - you don't want it to be something completely different.
A(A-1) = I or A-1(A) = I
There are a couple of exceptions, though. First of all, A-1 does not mean 1/A. Remember, "There is no Matrix Division!" Secondly, A-1 does not mean take the reciprocal of every element in the matrix A.
The matrix must be square (same number of rows and columns).
The determinant of the matrix must not be zero (determinants are covered in section 6.4). This is instead of the real number not being zero to have an inverse, the determinant must not be zero to have an inverse.
A square matrix that has an inverse is called invertible or non-singular. A matrix that does not have an inverse is called singular.
A matrix does not have to have an inverse, but if it does, the inverse is unique.
The inverse of a matrix A will satisfy the equation A(A-1) = I.
Adjoin the identity matrix onto the right of the original matrix, so that you have A on the left side and the identity matrix on the right side. It will look like this [ A | I ].
Row-reduce (I suggest using pivoting) the matrix until the left side is the Identity matrix. When the left side is the Identity matrix, the right side will be the Inverse [ I | A-1 ]. If you are unable to obtain the identity matrix on the left side, then the matrix is singular and has no inverse.
Take the augmented matrix from the right side and call that the inverse.
The inverse of a 2×2 matrix can be found by ...
Switch the elements on the main diagonal
Take the opposite of the other two elements
Divide all the values by the determinant of the matrix (since we haven't talked about the determinant, for a 2×2 system, it is the product of the elements on the main diagonal minus the product of the other two elements).
Example for the shortcut
Let's go with an original matrix of
Step 1, switch the elements on the main diagonal would involve switching the 5 and 7.
Step 2, take the opposite of the other two elements, but leave them where they are.
Step 3, find the determinant and divide every element by that. The determinant is the product of the elements on the main diagonal minus the product of the elements off the main diagonal. That means the determinant of this matrix is 7(5) - (-3)(2) = 35 + 6 = 41. We divide every element by 41.
The inverse of the original matrix is ...
Now, you're saying, wait a minute - you said there was no matrix division. There is no division by a matrix. You may multiply or divide a matrix by a scalar (real number) and the determinant is a scalar.