Suppose A = QR, where Q and R are n × n, R is invertible and upper triangular, and Q has the property that Q^T Q = I. Show that for each b in R^n, the equation Ax = b has a unique solution. What computations with Q and R will produce the solution?
Since Q is square and Q^T Q = I, Q is invertible by the Invertible Matrix Theorem with its inverse is Q^-1 = Q^T. Thus, A is the product of invertible matrices and hence is invertible and will be n × n. Thus, by Theorem 5, the equation Ax = b has a unique solution for all b. The equation Ax = b is equivalent to b, which is equivalent to which reduces to and finally x = . A good algorithm for finding x is to compute and then row reduce the matrix. The reduction is fast in this case because R is a triangular matrix.