Exercises 42–44 show how to use the condition number of a matrix A to estimate the accur...

Exercises 42–44 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.

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

Exercise 9:

Unless otherwise specified, assume that all matrices in these exercises are n × n. Determine which of the matrices in Exercises 1–10 are invertible. Use as few calculations as possible. Justify your answers.