91Ó°ÊÓ

Consider square matrices in which the entries are consecutive integers. An example of such a matrix is \(\left[ \begin{array}{r} 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{array} \right]\). (a) Use a graphing utility to evaluate the determinants of four matrices of this type. Make a conjecture based on the results. (b) Verify your conjecture.

PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices. \(\dfrac{8x^2}{(x-1)^2(x+1)} = \dfrac{A}{x+1} + \dfrac{B}{x-1} + \dfrac{C}{(x-1)^2}\)

FINANCE A small shoe corporation borrowed \(\$ 1,500,000\) to expand its line of shoes. Some of the money was borrowed at \(7 \%,\) some at \(8 \%,\) and some at 10\(\% .\) Use a system of equations to determine how much was borrowed at each rate if the annual interest was \(\$ 130,500\) and the amount borrowed at 10\(\%\) was 4 times the amount borrowed at 7\(\% .\) Solve the system using matrices.

TIPS A food server examines the amount of money earned in tips after working an 8-hour shift. The server has a total of \(\$95\) in denominations of \(\$1\), \(\$5\), \(\$10\), and \(\$20\) bills. The total number of paper bills is 26. The number of \(\$5\) bills is 4 times the number of \(\$10\) bills, and the number of \(\$1\) bills is 1 less than twice the number of \(\$5\) bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination.