B.2. Matrix inverse

The matrix inverse comes up a lot when you’re manipulating algebraic equations of matrices. The matrix X is the inverse of matrix Y if XY=I where I is the identity matrix. (The identity matrix often written as I is a matrix that’s all 0s except for the diagonals, which are 1s. You can multiply other matrices by the identity matrix and get the original matrix.) The practical drawback to the matrix inverse is that it becomes messy for matrices larger than a few elements and is rarely computed by hand. It helps to know when you can’t take the inverse of a matrix. Knowing this will help you avoid making errors in your programs. You write the inverse of a matrix B as B^-1.

A matrix has to be square to be invertible. By square, I mean the number of rows and columns has to be equal. Even if the matrix is square, it may not be invertible. If a matrix is not invertible, we say that it’s singular or degenerate. A matrix can be singular if you can express one column as a linear combination of other columns. If you could do this, you could reduce a column in the matrix to all 0s. An example of such a matrix is shown in figure B.9. This becomes a problem when computing the inverse of the matrix, because you’ll try to divide by zero. I’ll show you this shortly.