Chapter 4

Determine whether the matrices are multiplicative inverses. $$ \left[\begin{array}{rrr}{1} & {2} & {-1} \\ {-1.5} & {-3} & {1.75} \\ {0} & {-1} & {0.5}\end{array}\right],\left[\begin{array}{rrr}{1} & {0} & {2} \\ {3} & {2} & {-1} \\ {6} & {4} & {0}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{-1} & {3} \\ {5} & {2}\end{array}\right] $$

Use matrices \(A, B, C,\) and \(D .\) Find each product, sum, or difference. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{ll}{5} & {1} \\ {0} & {2}\end{array}\right]$$ $$ 2 A-5 B $$

Find each sum or difference. $$ \left[\begin{array}{rr}{0.5} & {9.5} \\ {-3.5} & {5.5}\end{array}\right]-\left[\begin{array}{rr}{0.5} & {9.5} \\ {-3.5} & {5.5}\end{array}\right] $$

Refer to matrices \(A\) and \(B\) at the right. Identify each matrix element. \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1.5} & {3} \\ {7} & {-2}\end{array}\right] \quad B=\left[\begin{array}{lll}{6} & {-3} & {\frac{1}{2}}\end{array}\right]\) \(a_{31}\)

Write a system of equations for each augmented matrix. $$ \left[\begin{array}{rr|r}{5} & {1} & {-3} \\ {-2} & {2} & {4}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{-2} & {0} \\ {2} & {-1}\end{array}\right] $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{array}{l}{300 x-y=130} \\ {200 x+y=120}\end{array}\right. $$

Solve each matrix equation. Check your answers. $$ 3\left[\begin{array}{rr}{2} & {0} \\ {-1} & {5}\end{array}\right]-2 X=\left[\begin{array}{rr}{-10} & {5} \\ {0} & {17}\end{array}\right] $$

Find each sum or difference. $$ \left[\begin{array}{rr}{1.5} & {-1.9} \\ {0} & {4.6}\end{array}\right]-\left[\begin{array}{cc}{8.3} & {-3.2} \\ {2.1} & {5.6}\end{array}\right] $$

Refer to matrices \(A\) and \(B\) at the right. Identify each matrix element. \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1.5} & {3} \\ {7} & {-2}\end{array}\right] \quad B=\left[\begin{array}{lll}{6} & {-3} & {\frac{1}{2}}\end{array}\right]\) \(b_{13}\)

Write a system of equations for each augmented matrix. $$ \left[\begin{array}{rr|r}{-1} & {2} & {-6} \\ {1} & {1} & {7}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{5} & {3} \\ {-2} & {1}\end{array}\right] $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{array}{l}{x+5 y=-4} \\ {x+6 y=-5}\end{array}\right. $$

Solve each matrix equation. $$ \left[\begin{array}{ccc}{5} & {1} & {-4} \\ {2} & {-3} & {-5} \\ {7} & {2} & {-6}\end{array}\right] X=\left[\begin{array}{l}{5} \\ {2} \\\ {5}\end{array}\right] $$

Solve each matrix equation. Check your answers. $$ 5 X-\left[\begin{array}{rr}{1.5} & {-3.6} \\ {-0.3} & {2.8}\end{array}\right]=\left[\begin{array}{rr}{0.2} & {1.3} \\ {-5.6} & {1.7}\end{array}\right] $$

Solve each matrix equation. $$ \left[\begin{array}{rr}{1} & {2} \\ {2} & {1} \\ {-3} & {4}\end{array}\right]+X=\left[\begin{array}{rr}{5} & {-6} \\ {1} & {0} \\\ {8} & {5}\end{array}\right] $$

Refer to matrices \(A\) and \(B\) at the right. Identify each matrix element. \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1.5} & {3} \\ {7} & {-2}\end{array}\right] \quad B=\left[\begin{array}{lll}{6} & {-3} & {\frac{1}{2}}\end{array}\right]\) \(a_{32}\)

Write a system of equations for each augmented matrix. $$ \left[\begin{array}{rrr|r}{2} & {1} & {1} & {1} \\ {1} & {1} & {1} & {2} \\\ {1} & {-1} & {1} & {-2}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{ll}{5} & {2} \\ {1} & {3}\end{array}\right] $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{aligned} 2 x+3 y &=12 \\ x+2 y &=7 \end{aligned}\right. $$

Solve each matrix equation. $$ \left[\begin{array}{rrr}{6} & {10} & {-13} \\ {4} & {-2} & {7} \\ {0} & {9} & {-8}\end{array}\right] X=\left[\begin{array}{l}{84} \\ {18} \\\ {56}\end{array}\right] $$

Graph each figure and its image after reflection in the given line. \(\left[\begin{array}{ccc}{-1} & {0} & {5} \\ {-1} & {5} & {0}\end{array}\right] ; y\) -axis

Find each product. $$ \left[\begin{array}{rr}{-3} & {4} \\ {5} & {2}\end{array}\right]\left[\begin{array}{rr}{1} & {0} \\ {2} & {-3}\end{array}\right] $$

Solve each matrix equation. $$ \left[\begin{array}{rrr}{2} & {1} & {-1} \\ {0} & {2} & {1}\end{array}\right]-X=\left[\begin{array}{rrr}{11} & {3} & {-13} \\ {15} & {-9} & {8}\end{array}\right] $$

Refer to matrices \(A\) and \(B\) at the right. Identify each matrix element. \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1.5} & {3} \\ {7} & {-2}\end{array}\right] \quad B=\left[\begin{array}{lll}{6} & {-3} & {\frac{1}{2}}\end{array}\right]\) \(a_{12}\)

Use an augmented matrix to solve each system. $$ \left\\{\begin{array}{l}{2 x-2 y=15} \\ {4 x+4 y=10}\end{array}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{ll}{2} & {-1} \\ {5} & {-4}\end{array}\right] $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{aligned} 2 x+3 y &=5 \\ x+2 y &=6 \end{aligned}\right. $$

Find each product. $$ \left[\begin{array}{rr}{1} & {0} \\ {2} & {-3}\end{array}\right]\left[\begin{array}{rr}{-3} & {4} \\ {5} & {2}\end{array}\right] $$

Solve each matrix equation. $$ X-\left[\begin{array}{rr}{1} & {4} \\ {-2} & {3}\end{array}\right]=\left[\begin{array}{rr}{5} & {-2} \\ {1} & {0}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{-4} & {3} \\ {2} & {0}\end{array}\right] $$

Use an augmented matrix to solve each system. $$ \left\\{\begin{array}{l}{2 x-4 y=20} \\ {4 x+2 y=-20}\end{array}\right. $$

Find the coordinates of each image after reflection in the given line. $$ \left[\begin{array}{rrrr}{3} & {6} & {3} & {6} \\ {-3} & {3} & {3} & {-3}\end{array}\right] ; y=-x $$

Find each product. $$ \left[\begin{array}{rr}{0} & {2} \\ {-4} & {0}\end{array}\right]\left[\begin{array}{rr}{0} & {2} \\ {-4} & {0}\end{array}\right] $$

Solve each matrix equation. $$ X+\left[\begin{array}{rr}{6} & {1} \\ {-2} & {3}\end{array}\right]=\left[\begin{array}{rr}{2} & {0} \\ {-3} & {1}\end{array}\right] $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{2} & {-1} \\ {1} & {0}\end{array}\right] $$

Use an augmented matrix to solve each system. $$ \left\\{\begin{aligned} x+2 y &=3 \\ 4 x+2 y &=-6 \end{aligned}\right. $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{aligned} 9 y+2 z &=18 \\ 3 x+2 y+z &=5 \\ x-y &=-1 \end{aligned}\right. $$

If matrix \(A\) has an inverse, what must be true? \(\begin{array}{llll}{\text { 1. } A A^{-1}=I} & {\text { II. } A^{-1} A=I} & {\text { III. } A^{-1} I=A^{-1}}\end{array}\) \(\begin{array}{ll}{\text { A I only }} & {\text { B II only }}\end{array}\) C) I and II only \(\quad\) D I, II, and III

Find the coordinates of each image after reflection in the given line. $$ \left[\begin{array}{llll}{0} & {4} & {8} & {6} \\ {0} & {4} & {4} & {2}\end{array}\right] ; x-a x i s $$

Find each product. $$ \left[\begin{array}{ll}{-3} & {5}\end{array}\right]\left[\begin{array}{r}{-3} \\\ {5}\end{array}\right] $$

Determine whether the two matrices in each pair are equal. Justify your reasoning. $$ \left[\begin{array}{rr}{-2} & {3} \\ {5} & {0}\end{array}\right],\left[\begin{array}{cc}{2(-1)} & {2(1.5)} \\ {2(2.5)} & {2(0)}\end{array}\right] $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{ll}{2} & {3} \\ {1} & {1}\end{array}\right] $$

Use an augmented matrix to solve each system. $$ \left\\{\begin{aligned} x+5 y &=-25 \\ 5 x+y &=25 \end{aligned}\right. $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{aligned} 9 y+2 z &=14 \\ 3 x+2 y+z &=5 \\ x-y &=-1 \end{aligned}\right. $$

Cryptography Two members of the Hopewell High School math club share messages in code. They use the alphabet table from Example \(5 .\) a. One of the students has lost her encoding matrix. Luckily, she remembers that the decoding matrix is $$E^{-1}=\left[\begin{array}{ccc}{1} & {-1} & {0} \\\ {0} & {1} & {-1} \\ {0} & {0} &{1}\end{array}\right] \text { . Compute } E \text { to find the encoding matrix. }$$ b. Use your answer to part (a) to encode the message MATH IS COOL. c. Open-Ended Use the encoding matrix from part (a) to encode a short sentence. Use the decoding matrix to check your work.

Find the coordinates of each image after reflection in the given line. $$ \left[\begin{array}{lllll}{1} & {2} & {3} & {4} & {2.5} \\ {3} & {2} & {2} & {3} & {5}\end{array}\right] ; y=x $$

Use the table below $$\begin{array}{|c|c|c|c|c|c|}\hline 1998 & {1999} & {2000} & {2001} & {2002} & {2003} \\ \hline 847.0 & {938.9} & {942.5} & {881.9} & {803.3} & {745.9} \\\ \hline 0.5 & {2.5} & {3.3} & {7.9} & {10.7} & {17.5} \\\ \hline\end{array}$$ Display the data in a matrix \(A\) with columns representing years. Identify \(a_{23}\) and tell what it represents.

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{ll}{2} & {3} \\ {2} & {4}\end{array}\right] $$