Chapter 1.1, Problem 8CP

The Hilbert matrix is the $n\times n$ matrix whose ijth entry is $1/\left(i+j-1\right)$ . Let $A$ denote the $5\times 5$ Hilbert matrix. Its largest eigenvalue is about 1.567. Use the Bisection Method to decide how to change the upper left entry $A_{11}$ to make the largest eigenvalue of $A$ equal to $\pi$ . Determine $A_{11}$ within six correct decimal places. You may use the MATLAB commands hilb, pi, eig, and max to simplify your task.