Chapter 4

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rrr}{1} & {2} & {6} \\ {1} & {-1} & {0} \\ {1} & {0} & {2}\end{array}\right] $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ H F $$

State the dimensions of each matrix. Identify the indicated element. \(\left[\begin{array}{lll}{-4} & {8} & {12}\end{array}\right], a_{13}\)

Solve each matrix equation. If an equation cannot be solved, explain why. $$ \left[\begin{array}{ll}{5} & {-3} \\ {4} & {-2}\end{array}\right] X=\left[\begin{array}{c}{5} \\ {10}\end{array}\right] $$

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{0.5 x+1.5 y=7} \\ {2.5 x-3.5 y=-9}\end{array}\right. $$

Solve each matrix equation. If the coefficient matrix has no inverse, write \(n o\) unique solution. $$ \left[\begin{array}{rr}{2} & {-3} \\ {-4} & {6}\end{array}\right]\left[\begin{array}{l}{a} \\\ {b}\end{array}\right]=\left[\begin{array}{r}{1} \\ {-2}\end{array}\right] $$

Evaluate the determinant of each matrix. Describe any patterns. $$\left[\begin{array}{lll}{1} & {2} & {3} \\ {1} & {2} & {3} \\ {1} & {2} & {3}\end{array}\right] \quad \mathbf{b} \cdot\left[\begin{array}{rrr}{-1} & {-2} & {-3} \\ {-3} & {-2} & {-1} \\ {-1} & {-2} & {-3}\end{array}\right] \quad \mathbf{c} \cdot\left[\begin{array}{rrr}{1} & {2} & {3} \\ {2} & {3} & {1} \\ {1} & {2} & {3}\end{array}\right] \quad \mathbf{d} \cdot\left[\begin{array}{rrr}{-1} & {2} & {-3} \\ {-1} & {-3} & {-1} \\ {-1} & {2} & {-3}\end{array}\right]$$

Each matrix represents the vertices of a polygon. Translate each figure 5 units left and 1 unit up. Express your answer as a matrix. $$ \left[\begin{array}{cccc}{-3} & {-3} & {2} & {2} \\ {-2} & {-4} & {-2} & {-4}\end{array}\right] $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ G H $$

Solve each matrix equation for \(X\) $$ \left[\begin{array}{rrr}{1} & {2} & {-3} \\ {2} & {1} & {3}\end{array}\right]+X=\left[\begin{array}{rrr}{5} & {1} & {8} \\ {-6} & {0} & {5}\end{array}\right] $$

State the dimensions of each matrix. Identify the indicated element. \(\left[\begin{array}{r}{-5} \\ {4} \\ {3}\end{array}\right], a_{31}\)

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{-1.2 x-0.3 y=2.1} \\ {-0.2 x+0.8 y=4.6}\end{array}\right. $$

Solve each matrix equation. If the coefficient matrix has no inverse, write \(n o\) unique solution. $$ \left[\begin{array}{ll}{2} & {1} \\ {4} & {3}\end{array}\right]\left[\begin{array}{l}{x} \\\ {y}\end{array}\right]=\left[\begin{array}{l}{10} \\ {-2}\end{array}\right] $$

Each matrix represents the vertices of a polygon. Translate each figure 5 units left and 1 unit up. Express your answer as a matrix. $$ \left[\begin{array}{cccc}{-3} & {0} & {3} & {0} \\ {-9} & {-6} & {-9} & {-12}\end{array}\right] $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ H G $$

State the dimensions of each matrix. Identify the indicated element. \(\left[\begin{array}{rr}{-16} & {24} \\ {8} & {-2}\end{array}\right], a_{21}\)

Solve each matrix equation for \(X\) $$ X-\left[\begin{array}{cc}{4} & {12} \\ {75} & {-1}\end{array}\right]=\left[\begin{array}{rr}{5} & {50} \\ {50} & {-10}\end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rr}{4} & {5} \\ {-4} & {4}\end{array}\right| $$

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

Solve each system. $$ \left\\{\begin{aligned}-3 x+4 y &=2 \\ x-y &=-1 \end{aligned}\right. $$

Each matrix represents the vertices of a polygon. Translate each figure 5 units left and 1 unit up. Express your answer as a matrix. $$ \left[\begin{array}{rrr}{0} & {1} & {-4} \\ {0} & {3} & {5}\end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rr}{-3} & {10} \\ {6} & {20}\end{array}\right| $$

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{\frac{x}{2}+\frac{y}{4}=4} \\ {\frac{x}{4}-\frac{3 y}{8}=-2}\end{array}\right. $$

Solve each system. $$ \left\\{\begin{aligned} x+2 y &=10 \\ 3 x+5 y &=26 \end{aligned}\right. $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ J F $$

Evaluate each determinant. $$ \left|\begin{array}{cc}{-\frac{1}{2}} & {2} \\ {-2} & {8}\end{array}\right| $$

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{2 x+3 y+5 z=12} \\ {4 x+2 y+4 z=-2} \\ {5 x+4 y+7 z=7}\end{array}\right. $$

Solve each system. $$ \left\\{\begin{aligned} x-3 y &=-1 \\\\-6 x+19 y &=6 \end{aligned}\right. $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ H J $$

Writing Suppose \(A\) and \(B\) are two matrices with the same dimensions. a. Explain how to find \(A+B\) and \(A-B .\) b. Explain how to find a matrix \(C\) such that \(A+C=O .\)

Use an augmented matrix to solve each system. $$ \left\\{\begin{aligned} x+y+z &=1 \\ y-3 z &=4 \\ x &-z=2 \end{aligned}\right. $$

Evaluate each determinant. $$ \left|\begin{array}{ll}{2} & {0} \\ {0} & {1}\end{array}\right| $$

Solve each system. $$ \left\\{\begin{array}{l}{x=5-y} \\ {3 y=z} \\ {x+z=7}\end{array}\right. $$

What is the determinant of \(\left[\begin{array}{ccc}{5} & {0} & {0} \\ {0} & {5} & {0} \\ {0} & {0} & {5}\end{array}\right] ?\) \(\begin{array}{llll}{\text { A. } 5} & {\text { B. } 25} & {\text { C. } 125} & {\text { D. } 555}\end{array}\)

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ J H $$

Solve each equation for each variable. $$ \left[\begin{array}{ccc}{4 b+2} & {-3} & {4 d} \\ {-4 a} & {2} & {3} \\ {2 f-1} & {-14} & {1}\end{array}\right]=\left[\begin{array}{ccc}{11} & {2 c-1} & {0} \\ {-8} & {2} & {3} \\ {0} & {3 g-2} & {1}\end{array}\right] $$

Use an augmented matrix to solve each system. $$ \left\\{\begin{aligned} x+y+z &=0 \\ y+4 z &=-6 \\ 2 x &-2 z=& 4 \end{aligned}\right. $$

Evaluate each determinant. $$ \begin{array}{ll}{6} & {9} \\ {3} & {6}\end{array} | $$

Solve each system. $$ \left\\{\begin{array}{l}{-x=-4-z} \\ {2 y=z-1} \\\ {x=6-y-z}\end{array}\right. $$

Which matrix has no inverse? $$ \mathbf{F} \cdot\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {0} \\\ {0} & {0} & {1}\end{array}\right] \quad \text { G. }\left[\begin{array}{lll}{1} & {0} & {1} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ \mathrm{H.}\left[\begin{array}{rrr}{0} & {0} & {1} \\ {0} & {-1} & {0} \\\ {1} & {1} & {0}\end{array}\right] \quad \text { I. }\left[\begin{array}{rrr}{1} & {-1} & {0} \\ {0} & {0} & {1} \\ {-1} & {0} & {1}\end{array}\right] $$

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

Which element in matrix \(A=\left[\begin{array}{llll}{5} & {-2} & {-\frac{1}{2}} & {0}\end{array}\right]\) is the number 0\(?\) A. \(a_{40}\) B. \(a_{4}\) C. \(a_{14}\) D. \(a_{41}\)

Solve each equation for each variable. $$ \left[\begin{array}{cc}{x^{2}} & {4} \\ {-2} & {y^{2}}\end{array}\right]=\left[\begin{array}{rr}{9} & {4} \\ {-2} & {5 y}\end{array}\right] $$

Use an augmented matrix to solve each system. $$ \left\\{\begin{array}{rr}{2 x+y} & {=8} \\ {x+z=} & {5} \\ {y-z} & {=-1}\end{array}\right. $$

Determine whether the matrices are multiplicative inverses. If they are not, explain why not. $$ \left[\begin{array}{cc}{2} & {0.5} \\ {5} & {1}\end{array}\right],\left[\begin{array}{cc}{-2} & {1} \\ {10} & {-4}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned}-b+2 c &=4 \\ a+b-c &=-10 \\ 2 a+& 3 c=1 \end{aligned}\right. $$

What is the determinant of the identity matrix \(/ ?\)

Mental Math Find each product. $$ -1\left[\begin{array}{rrr}{9} & {-7} & {-4} \\ {-8} & {-2} & {3}\end{array}\right] $$

In which matrix is the value of \(a_{32}\) less than the value of \(a_{21} ?\) F. \(\left[\begin{array}{rrr}{-1} & {0} & {5} \\ {4} & {3} & {-1} \\ {-3} & {2} & {6}\end{array}\right]\) G. \(\left[\begin{array}{rrr}{-1} & {5} & {0} \\ {3} & {4} & {-1} \\ {-3} & {6} & {2}\end{array}\right]\) H. \(\left[\begin{array}{rrr}{0} & {5} & {-1} \\ {-1} & {4} & {3} \\ {6} & {2} & {-3}\end{array}\right]\) J. \(\left[\begin{array}{rrr}{0} & {-1} & {5} \\ {0} & {3} & {4} \\ {-3} & {1} & {6}\end{array}\right]\)

Solve each equation for each variable. $$ \left[\begin{array}{ccc}{4 c} & {2-d} & {5} \\ {-3} & {-1} & {2} \\ {0} & {-10} & {15}\end{array}\right]=\left[\begin{array}{ccc}{2 c+5} & {4 d} & {g} \\\ {-3} & {h} & {f-g} \\ {0} & {-4 c} & {15}\end{array}\right] $$