Inverse matrices

Let's revisit the concept of inverse matrices and go a little more in depth with them. We know from earlier that AA^-1 = I, but not every matrix has an inverse. 

There are, again, some rules we must follow when it comes to finding the inverses of matrices, as follows:

• The inverse only exists if, through the process of upper or lower triangular factorization, we obtain all the pivot values on the diagonal.
• If the matrix is invertible, it has only one unique inverse matrix—that is, if AB = I and AC = I, then B = C.
• If A is invertible, then to solve Av = b we multiply both sides by A^-1 and get AA^-1v = A^-1b, which finally gives us = A^-1b.
• If v is nonzero and b = 0, then the matrix does not have an inverse.
• 2 x 2 matrices are invertible only if ad - bc ≠ 0, where the following applies:

And ad - bc is called the determinant of A. A^-1 involves dividing each element in the matrix by the determinant.

• Lastly, if the matrix has any zero values along the diagonal, it is non-invertible.

Sometimes, we may have to invert the product of two matrices, but that is only possible when both the matrices are individually invertible (follow the rules outlined previously). 

For example, let's take two matrices A and B, which are both invertible. Then,  so that .