Chapter 4

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{2 x+y=4} \\ {3 x-y=6}\end{array}\right. $$

Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix. $$ \left\\{\begin{array}{l}{x+y=5} \\ {x-2 y=-4}\end{array}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{lll}{1} & {2} & {5} \\ {3} & {1} & {0} \\ {1} & {2} & {1}\end{array}\right] $$

Show that the matrices are multiplicative inverses. $$ \left[\begin{array}{rr}{3} & {2} \\ {4} & {3}\end{array}\right],\left[\begin{array}{rr}{3} & {-2} \\ {-4} & {3}\end{array}\right] $$

Use matrix addition to find the coordinates of each image after a translation of 3 units left and 5 units up. If possible, graph each pair of figures on the same coordinate plane. $$A(1,-3), B(1,1), C(5,1), D(5,-3)$$

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

State the dimensions of each matrix. \(\left[\begin{array}{rrr}{4} & {-2} & {2} \\ {1} & {4} & {1} \\ {0} & {5} & {-7}\end{array}\right]\)

Use Cramer's Rule to solve each system. $$ \left\\{\begin{aligned} 2 x+y &=7 \\\\-2 x+5 y &=-1 \end{aligned}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{lll}{1} & {4} & {0} \\ {2} & {3} & {5} \\ {0} & {1} & {0}\end{array}\right] $$

Use matrix addition to find the coordinates of each image after a translation of 3 units left and 5 units up. If possible, graph each pair of figures on the same coordinate plane. $$ G(0,0), H(4,4), I(4,-4), J(8,0) $$

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

Find each sum or difference. $$ \left[\begin{array}{rrr}{2} & {-3} & {4} \\ {5} & {6} & {-7}\end{array}\right]+\left[\begin{array}{ccc}{0} & {0} & {0} \\ {0} & {0} & {0}\end{array}\right] $$

State the dimensions of each matrix. $$\left[\begin{array}{r}1 \\\\-9 \\\5\end{array}\right]$$

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{2 x+4 y=10} \\ {3 x+5 y=14}\end{array}\right. $$

Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix. $$ \left\\{\begin{aligned} 3 a+5 b &=0 \\ a+b &=2 \end{aligned}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{-2} & {4} & {1} \\ {3} & {0} & {-1} \\ {1} & {2} & {1}\end{array}\right] $$

Show that the matrices are multiplicative inverses. $$ \left[\begin{array}{rr}{\frac{1}{5}} & {-\frac{1}{10}} \\ {0} & {\frac{1}{4}}\end{array}\right],\left[\begin{array}{ll}{5} & {2} \\ {0} & {4}\end{array}\right] $$

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

Find each sum or difference. $$ \left[\begin{array}{ll}{1} & {3} \\ {4} & {0}\end{array}\right]+\left[\begin{array}{rr}{0} & {5} \\ {-1} & {2}\end{array}\right]+\left[\begin{array}{ll}{0} & {-5} \\ {1} & {-2}\end{array}\right] $$

State the dimensions of each matrix. \(\left[\begin{array}{ll}{2} & {\sqrt{5}}\end{array}\right]\)

Use Cramer's Rule to solve each system. $$ \left\\{\begin{aligned} y+4 z &=5 \\ x+y+z &=8 \\ 2 x-5 y &=7 \end{aligned}\right. $$

Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix. $$ \left\\{\begin{aligned} x+3 y-z &=2 \\ x &+2 z=8 \\ 2 y-z &=1 \end{aligned}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{7} & {2} \\ {0} & {-3}\end{array}\right] $$

Use matrix addition to find the coordinates of each image after a translation of 3 units left and 5 units up. If possible, graph each pair of figures on the same coordinate plane. $$ R(9,3), S(3,6), T(3,3), U(6,-3) $$

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

Find each sum or difference. $$ \left[\begin{array}{rr}{6.4} & {-1.9} \\ {-6.4} & {0.8}\end{array}\right]+\left[\begin{array}{rr}{-2.5} & {-0.4} \\ {5.8} & {8.3}\end{array}\right] $$

State the dimensions of each matrix. \(\left[\begin{array}{ccc}{3} & {2} & {1} \\ {2} & {0} & {-3}\end{array}\right]\)

Use Cramer's Rule to solve each system. $$ \left\\{\begin{aligned} 2 x+3 y+z &=5 \\ x+y-2 z &=-2 \\\\-3 x &+z=-7 \end{aligned}\right. $$

Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix. $$ \left\\{\begin{aligned} r-s+t &=150 \\ 2 r+t &=425 \\ s+3 t &=0 \end{aligned}\right. $$

Use a graphing calculator to evaluate the determinant of each \(3 \times 3\) matrix. $$ \left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{6} & {2} \\ {-6} & {-2}\end{array}\right] $$

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

Find each sum or difference. $$ \left[\begin{array}{rr}{6} & {-3} \\ {-7} & {2}\end{array}\right]+\left[\begin{array}{rr}{-6} & {3} \\ {7} & {-2}\end{array}\right] $$

State the dimensions of each matrix. \(\left[\begin{array}{r}{2.5} \\ {-3} \\ {-1.6} \\ {10.0}\end{array}\right]\)

Write an augmented matrix for each system. $$ \left\\{\begin{array}{l}{3 x-4 y=17} \\ {8 x+y=-3}\end{array}\right. $$

Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix. $$ \left\\{\begin{array}{c}{x+2 y=11} \\ {2 x+3 y=18}\end{array}\right. $$

Use a graphing calculator to evaluate the determinant of each \(3 \times 3\) matrix. $$ \left[\begin{array}{rrr}{0} & {-2} & {-3} \\ {1} & {2} & {4} \\ {-2} & {0} & {1}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{cc}{0} & {0.5} \\ {1.5} & {2}\end{array}\right] $$

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

Find each sum or difference. $$ \left[\begin{array}{rrr}{5} & {4} & {3} \\ {1} & {-2} & {6}\end{array}\right]-\left[\begin{array}{lll}{1} & {1} & {1} \\ {1} & {1} & {1}\end{array}\right] $$

Refer to matrices \(A\) and \(B\) at the right. Identify each matrix element. \(A=\left[\begin{array}{rr}{0} & {-1} \\ {1.5} & {3} \\ {7} & {-2}\end{array}\right] \quad B=\left[\begin{array}{lll}{6} & {-3} & {\frac{1}{2}}\end{array}\right]\) \(a_{21}\)

Write an augmented matrix for each system. $$ \left\\{\begin{aligned} 3 x-7 y+3 z &=-3 \\ x+y+2 z &=-3 \\ 2 x-3 y+5 z &=-8 \end{aligned}\right. $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{array}{l}{x+3 y=5} \\ {x+4 y=6}\end{array}\right. $$

Use a graphing calculator to evaluate the determinant of each \(3 \times 3\) matrix. $$ \left[\begin{array}{ccc}{12.2} & {13.3} & {9} \\ {1} & {-4} & {-17} \\\ {21.4} & {-15} & {0}\end{array}\right] $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{cc}{\frac{1}{2}} & {\frac{2}{3}} \\ {\frac{3}{5}} & {\frac{1}{4}}\end{array}\right] $$

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

Find each sum or difference. $$ \left[\begin{array}{lll}{2} & {1} & {2} \\ {1} & {2} & {1}\end{array}\right]-\left[\begin{array}{lll}{2} & {3} & {2} \\ {3} & {2} & {3}\end{array}\right] $$

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

Write an augmented matrix for each system. $$ \left\\{\begin{aligned}-x+5 y &=-1 \\ x-2 y &=1 \end{aligned}\right. $$

Solve each system of equations. Check your answers. $$ \left\\{\begin{aligned} p-3 q &=-1 \\\\-5 p+16 q &=5 \end{aligned}\right. $$