Chapter 9

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{aligned} 2 x+y+2 z &=2 \\ 3 x-5 y-z &=4 \\ x-2 y-3 z &=-6 \end{aligned}\right. $$

a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23},\) or explain why identification is not possible. $$ \left[\begin{array}{rrr} {4} & {-7} & {5} \\ {-6} & {8} & {-1} \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{ll} {5} & {7} \\ {2} & {3} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} {4} & {-3} \\ {-5} & {4} \end{array}\right], \quad B=\left[\begin{array}{ll} {4} & {3} \\ {5} & {4} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} 5 x+12 y+z &=10 \\ 2 x+5 y+2 z &=-1 \\ x+2 y-3 z &=5 \end{aligned}\right. $$

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{aligned} 3 x-2 y+5 z &=31 \\ x+3 y-3 z &=-12 \\ -2 x-5 y+3 z &=11 \end{aligned}\right. $$

Evaluate each determinant. $$ \left|\begin{array}{ll} {4} & {8} \\ {5} & {6} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} {-2} & {-1} \\ {-1} & {1} \end{array}\right], \quad B=\left[\begin{array}{ll} {1} & {1} \\ {1} & {2} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} 2 x-4 y+z &=3 \\ x-3 y+z &=5 \\ 3 x-7 y+2 z &=12 \end{aligned}\right. $$

a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23},\) or explain why identification is not possible. $$ \left[\begin{array}{rrr} {-6} & {4} & {-1} \\ {-9} & {0} & {\frac{1}{2}} \end{array}\right] $$

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{aligned} x-y+z &=8 \\ y-12 z &=-15 \\ z &=1 \end{aligned}\right. $$

a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23},\) or explain why identification is not possible. $$ \left[\begin{array}{rrrr} {1} & {-5} & {\pi} & {e} \\ {0} & {7} & {-6} & {-\pi} \\ {-2} & {\frac{1}{2}} & {11} & {-\frac{1}{5}} \end{array}\right] $$

Evaluate each determinant. $$Therefore, the given determinant evaluates to -29 \left|\begin{array}{rr} {-4} & {1} \\ {5} & {6} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} {-4} & {0} \\ {1} & {3} \end{array}\right], \quad B=\left[\begin{array}{rr} {-2} & {4} \\ {0} & {1} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} 5 x+8 y-6 z &=14 \\ 3 x+4 y-2 z &=8 \\ x+2 y-2 z &=3 \end{aligned}\right. $$

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{array}{r} {x-2 y+3 z=9} \\ {y+3 z=5} \\ {z=2} \end{array}\right. $$

a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23},\) or explain why identification is not possible. $$ \left[\begin{array}{rrrr} {-4} & {1} & {3} & {-5} \\ {2} & {-1} & {\pi} & {0} \\ {1} & {0} & {-e} & {\frac{1}{5}} \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rr} {7} & {9} \\ {-2} & {-5} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} {-2} & {4} \\ {1} & {-2} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {2} \\ {-1} & {-2} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{array}{c} {5 x-11 y+6 z=12} \\ {-x+3 y-2 z=-4} \\ {3 x-5 y+2 z=4} \end{array}\right. $$

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{array}{r} {5 x-2 y-3 z=0} \\ {x+y=5} \\ {2 x-3 z=4} \end{array}\right. $$

Find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{l} {x} \\ {4} \end{array}\right]=\left[\begin{array}{l} {6} \\ {y} \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rr} {-7} & {14} \\ {2} & {-4} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rr} {-2} & {1} \\ {\frac{1}{2}} & {-\frac{1}{2}} \end{array}\right], \quad B=\left[\begin{array}{ll} {1} & {2} \\ {3} & {4} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{array}{r} {3 x+4 y+2 z=3} \\ {4 x-2 y-8 z=-4} \\ {x+y-z=3} \end{array}\right. $$

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{aligned} x-2 y+z &=10 \\ 3 x+y &=5 \\ 7 x+2 z &=2 \end{aligned}\right. $$

Find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{l} {x} \\ {7} \end{array}\right]=\left[\begin{array}{l} {11} \\ {y} \end{array}\right] $$

Evaluate each determinant.Therefore, the given determinant evaluates to -22 $$ \left|\begin{array}{rr} {1} & {-3} \\ {-8} & {2} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{ll} {4} & {5} \\ {2} & {3} \end{array}\right], \quad B=\left[\begin{array}{rr} {\frac{1}{2}} & {-\frac{5}{2}} \\ {-1} & {2} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} 2 x-y-z &=0 \\ x+2 y+z &=3 \\ 3 x+4 y+2 z &=8 \end{aligned}\right. $$

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{aligned} 2 w+5 x-3 y+z &=2 \\ 3 x+y &=4 \\ w-x+5 y &=9 \\ 5 w-5 x-2 y &=1 \end{aligned}\right. $$

Find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{rr} {x} & {2 y} \\ {z} & {9} \end{array}\right]=\left[\begin{array}{rr} {4} & {12} \\ {3} & {9} \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{cc} {-5} & {-1} \\ {-2} & {-7} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{lll} {0} & {1} & {0} \\ {0} & {0} & {1} \\ {1} & {0} & {0} \end{array}\right], \quad B=\left[\begin{array}{lll} {0} & {0} & {1} \\ {1} & {0} & {0} \\ {0} & {1} & {0} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} 8 x+5 y+11 z &=30 \\ -x-4 y+2 z &=3 \\ 2 x-y+5 z &=12 \end{aligned}\right. $$

Write the augmented matrix for each system of linear equations. $$ \left\\{\begin{aligned} 4 w+7 x-8 y+z &=3 \\ 5 x+y &=5 \\ w-x-y &=17 \\ 2 w-2 x+11 y &=4 \end{aligned}\right. $$

Find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{rr} {x} & {y+3} \\ {2 z} & {8} \end{array}\right]=\left[\begin{array}{rr} {12} & {5} \\ {6} & {8} \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rr} {\frac{1}{5}} & {\frac{1}{6}} \\ {-6} & {5} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rrr} {-2} & {1} & {-1} \\ {-5} & {2} & {-1} \\ {3} & {-1} & {1} \end{array}\right], \quad B=\left[\begin{array}{rrr} {1} & {0} & {1} \\ {2} & {1} & {3} \\ {-1} & {1} & {1} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} x+y-10 z &=-4 \\ x &-7 z=-5 \\ 3 x+5 y-36 z &=-10 \end{aligned}\right. $$

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{ll} {4} & {1} \\ {3} & {2} \end{array}\right], \quad B=\left[\begin{array}{ll} {5} & {9} \\ {0} & {7} \end{array}\right] $$

Write the system of linear equations represented by the augmented matrix. Use \(x, y,\) and \(z,\) or, if necessary, \(w, x, y\) and \(z,\) for the variables. $$ \left[\begin{array}{rrr|r} {5} & {0} & {3} & {-11} \\ {0} & {1} & {-4} & {12} \\ {7} & {2} & {0} & {3} \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{cc} {\frac{1}{2}} & {\frac{1}{2}} \\ {\frac{1}{8}} & {-\frac{3}{4}} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{lll} {1} & {2} & {3} \\ {1} & {3} & {4} \\ {1} & {4} & {3} \end{array}\right], \quad B=\left[\begin{array}{rrr} {\frac{7}{2}} & {-3} & {\frac{1}{2}} \\ {-\frac{1}{2}} & {0} & {\frac{1}{2}} \\ {-\frac{1}{2}} & {1} & {-\frac{1}{2}} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} w-2 x-y-3 z &=-9 \\ w+x-y &=0 \\ 3 w+4 x &+z=6 \\ 2 x-2 y+z &=3 \end{aligned}\right. $$

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rr} {-2} & {3} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{ll} {8} & {1} \\ {5} & {4} \end{array}\right] $$

Write the system of linear equations represented by the augmented matrix. Use \(x, y,\) and \(z,\) or, if necessary, \(w, x, y\) and \(z,\) for the variables. $$ \left[\begin{array}{rrr|r} {7} & {0} & {4} & {-13} \\ {0} & {1} & {-5} & {11} \\ {2} & {7} & {0} & {6} \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rr} {\frac{2}{3}} & {\frac{1}{3}} \\ {-\frac{1}{2}} & {\frac{3}{4}} \end{array}\right| $$

Find the products AB and BA to determine.whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{lll} {0} & {2} & {0} \\ {3} & {3} & {2} \\ {2} & {5} & {1} \end{array}\right], \quad B=\left[\begin{array}{rrr} {-3.5} & {-1} & {2} \\ {0.5} & {0} & {0} \\ {4.5} & {2} & {-3} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{aligned} 2 w+x-2 y-z &=3 \\ w-2 x+y+z &=4 \\ -w-8 x+7 y+5 z &=13 \\ 3 w+x-2 y+2 z &=6 \end{aligned}\right. $$