The Inverse of a Square Matrix,  Requirements, Finding the Inverse, Examples

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.

The Inverse of a Matrix

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.

Requirements to have an Inverse

• 1. The matrix must be square (same number of rows and columns).

  2. 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.

Finding the Inverse the Hard Way

The inverse of a matrix A will satisfy the equation A(A-1) = I.

• 1. 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 ].

  2. 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.

  3. Take the augmented matrix from the right side and call that the inverse.

Shortcut to the Finding the Inverse of a 2×2 Matrix

The inverse of a 2×2 matrix can be found by ...

• 1. Switch the elements on the main diagonal

  2. Take the opposite of the other two elements

  3. 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