## Properties of

(i) If a square matrix  satisfies either or ,'' then is invertible, and has the unique inverse .

(ii) When and are invertible, we have the following properties:

(iii) If is invertible, the equation  has the unique solution  .

The function inv(A) computes the inverse of a matrix A, if is invertible. Then the solution to  is given by inv(A) * b. Besides, Matlab/Octave has the special operator \'', called left division, which computes the solution immediately by A \ b. In summary the following two commands give the same solution:

> inv(A) * b
> A \ b


EXAMPLES 1. Let and . Then show that .

EXAMPLES 2. Find the inverse of

EXAMPLES 3. Solve the system

Department of Mathematics