The Inverse of a Matrix (10/14/05) If A is a square (say n by n) matrix and if there is an n by n matrix C such that C A = A C = I n, then C is called the inverse of A, is denoted A -1, and A is said to be an invertible matrix. Viewed as functions from R n to R n, this says that A and A -1 are inverse functions of each other (since I n is the identity function).

The 2 by 2 Case If A = then the determinant of A is the number a d – b c. This is denoted det(A). It is easy to check that: If det(A)  0 then the inverse of A is 1/det(A) If det(A) = 0, then A is not invertible.

Using Inverses If A is an invertible (hence square) matrix and if we wish to solve the matrix equation A x = b (x and b vectors in R n ), then x = A -1 b. That is, you can either solve the equation by our standard method, or you can compute A -1 and multiply b by it. In general, the former way is less work (but not always).

Computing Inverses An algorithm for computing A -1 is: Write down the n by 2n augmented matrix [A I n ] Do row reduction until the A half is in reduced echelon form. If the left half is now I n, then the matrix on the right is A -1. If the left half is not I n (i.e., not a pivot in every row/column), then A is not invertible.

The Invertible Matrix Theorem See Page 129 of our text. The theorem is a list of equivalent ways of saying that a given matrix A is invertible. It basically boils down to the fact that A is invertible if and only if as a linear transformation from R n to R n A is one-to-one (and hence also onto).

Assignment for Monday Catch up on any problems you have left behind in Chapter 2 so far. Read Section 2.3. Do the Practice and Exercises 1–7 odd, 11, 13 and 19.