Hands-On Mathematics for Deep Learning

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 ...