Chapter 10, Problem 24P

(a) Create a $3\times 3$ Hilbert matrix. This will be your matrix $\left[A\right]$. Multiply the matrix by the column vector $\left\{x\right\}={\left[1,1,1\right]}^T$. The solution of $\left[A\right]\left\{x\right\}$ will be another column vector $\left\{b\right\}$. Using any numerical package and Gauss elimination, find the solution to $\left[A\right]\left\{x\right\}=\left\{b\right\}$ using the Hilbert matrix and the vector $\left\{b\right\}$ that you calculated. Compare the result to your known $\left\{x\right\}$ vector. Use sufficient precision in displaying results to allow you to detect imprecision.

(b) Repeat part (a) using a $7\times 7$ Hilbert matrix.

(c) Repeat part (a) using a $10\times 10$ Hilbert matrix.