Inverse of an Invertible Matrix

What is (A ^. B)^-1 where A and B are n x n matrices?

B^{-1 .} A^-1 (order is reversed)

What is the Gaussian elimination algorithm?

• Consider the augmented matrix with dimensions n x 2n, (A | In) where A is an n x n matrix

• Perform EROs on the matrix to reduce the n x n block on the left to reduced echelon form (denoted REch(A)) i.e. (A | In) → (REch(A) | B)

• If REch(A) = In then B = A^-1. If REch(A) =/ In then A is not invertible

How can you tell early if a matrix is invertible?

You get a row of 0s whole performing Gaussian elimination algorithm

What is an elementary matrix?

Any matrix obtained by performing one ERO to the identity matrix I