Show how to use the condition number of a matrix A to estimate the accuracy of a computed solution of Ax = b. If the entries of A and b are accurate to about r significant digits and if the condition number of A is approximately 10^k (with k a positive integer), then the computed solution of Ax = b should usually be accurate to at least r - k significant digits.

Find the condition number of the matrix A in Exercise 9. Construct a random vector x in R⁴ and compute b = Ax. Then use your matrix program to compute the solution xj of Ax = b. To how many digits do x and x₁ agree? Find out the number of digits your matrix program stores accurately, and report how many digits of accuracy are lost when x₁ is used in place of the exact solution x.

Data from in Exercise 9

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4 -6 7 - 1 0 1 -5 2 -7 11 10 3 -7 9 19 -1