Chapter 2.5, Problem 66E

Solve each exercise by using the inverse of the coefficient matrix to solve a system of equations.

Music During a marching band's half-time show, the band members generally line up in such a way that a common shape is recognized by the fans. For example, as illustrated in the figure, a band might form a letter T, where an x represents a member of the band. As the music is played, the band will either create a new shape or rotate the original shape. In doing this, each member of the band will need to move from one point on the field to another. For larger bands, keeping track of who goes where can be a daunting task. However, it is possible to use matrix inverses to make the process a bit easier. The entire process is calculated by knowing how three band members, all of whom cannot be in a straight line, will move from the current position to a new position. For example, in the figure, we can see that there are band members at (50, 0), (50, 15), and (45, 20). We will assume that these three band members move to (40, 10), (55, 10), and (60, 15), respectively. (Note: The x-coordinate 60 is represented by the right side's 40-yard line in the picture.) Source: The College Mathematics Journal

(a) Find the inverse of $B=\left[\begin{array}{l}\begin{array}{ccc}50& 50& 45\end{array}\\ \begin{array}{ccc}0& 15& 20\end{array}\\ \begin{array}{ccc}1& 1& 1\end{array}\end{array}\right].$

(b) Find $A=\left[\begin{array}{l}\begin{array}{ccc}40& 55& 60\end{array}\\ \begin{array}{ccc}10& 10& 15\end{array}\\ \begin{array}{ccc}1& 1& 1\end{array}\end{array}\right]B^{-1.}$

(c) Use the result of part (b) to find the new position of the other band members. What is the shape of the new position? (Hint: Multiply the matrix A by a 3 × 1 column vector with the first two components equal to the original position of each band member and the third component equal to 1. The new position of the band member is in the first two components of the product.)