Chapter 6

In Exercises \(1-4\) 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. $$ \left|\begin{array}{ll}5 & 7 \\\2 & 3\end{array}\right| $$

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

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

In Exercises \(1-8,\) write the augmented matrix for each system of linear equations. \(\begin{array}{rr}2 x+y+2 z= & 2 \\ 3 x-5 y-z= & 4 \\ x-2 y-3 z= & -6\end{array}\)

In Exercises \(1-4\) 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. $$ \left[\begin{array}{rrr} -6 & 4 & -1 \\ -9 & 0 & \frac{1}{2} \end{array}\right] $$

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

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

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

In Exercises \(1-8,\) write the augmented matrix for each system of linear equations. \(\begin{aligned} 3 x-2 y+5 z &=31 \\ x+3 y-3 z &=-12 \\\\-2 x-5 y+3 z &=11 \end{aligned}\)

In Exercises \(1-4\) 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. $$ \left[\begin{array}{rrrr} 1 & -5 & \pi & e \\ 0 & 7 & -6 & -\pi \\ -2 & 1 & 11 & -1 \end{array}\right] $$

Evaluate each determinant. $$ \left|\begin{array}{rr}-4 & 1 \\\5 & 6\end{array}\right| $$

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

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

\(\begin{aligned} x-y+z &=8 \\ y-12 z &=-15 \\ z &=1 \end{aligned}\)

In Exercises \(1-4\) 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. $$ \left|\begin{array}{rr}7 & 9 \\\\-2 & -5\end{array}\right| $$

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

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

In Exercises \(1-8,\) write the augmented matrix for each system of linear equations. \(\begin{aligned} x-2 y+3 z &=9 \\ y+3 z &=5 \\ z &=2 \end{aligned}\)

In Exercises \(5-8,\) 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. $$ \left|\begin{array}{rr}-7 & 14 \\\2 & -4\end{array}\right| $$

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

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

In Exercises \(1-8,\) write the augmented matrix for each system of linear equations. \(\begin{aligned} 5 x-2 y-3 z &=0 \\ x+y &=5 \\ 2 x-3 z &=4 \end{aligned}\)

In Exercises \(5-8,\) find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{l} x \\ 7 \end{array}\right]=\left[\begin{array}{c} 11 \\ y \end{array}\right] $$

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

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

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

In Exercises \(1-8,\) write the augmented matrix for each system of linear equations. \(\begin{aligned} x-2 y+z &=10 \\ 3 x+y &=5 \\ 7 x+2 z &=2 \end{aligned}\)

In Exercises \(5-8,\) 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. $$ \left|\begin{array}{ll}-5 & -1 \\\\-2 & -7\end{array}\right| $$

find the products \(A B\) and \(B A\) 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. $$\begin{array}{l}8 x+5 y+11 z=30 \\\\-x-4 y+2 z=3 \\\2 x-y+5 z=12\end{array}$$

In Exercises \(1-8,\) write the augmented matrix for each system of linear equations. \(\begin{aligned} 2 w+5 x-3 y+z &=2 \\ 3 x+y &=4 \\ w-x+5 y &=9 \\ 5 w-5 x-2 y &=1 \end{aligned}\)

In Exercises \(5-8,\) 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 \(A B\) and \(B A\) 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. $$\begin{aligned}x+y-10 z &=-4 \\\x &-7 z=-5 \\\3 x+5 y-36 z &=-10\end{aligned}$$

In Exercises \(1-8,\) write the augmented matrix for each system of linear equations. \(\begin{aligned} 4 w+7 x-8 y+z &=3 \\ 5 x+y &=5 \\ w-x-y &=17 \\ 2 w-2 x+11 y &=4 \end{aligned}\)

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{ll} 4 & 1 \\ 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 5 & 9 \\ 0 & 7 \end{array}\right] $$

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

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

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

In Exercises \(9-12,\) write the system of linear equations represented by the augmented matrix. Use \(x, y, z,\) and, if necessary, \(w, x, y,\) and \(z,\) for the variables. \(\left[\begin{array}{rrr|r}5 & 0 & 3 & -11 \\ 0 & 1 & -4 & 12 \\ 7 & 2 & 0 & 3\end{array}\right]\)

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rr} -2 & 3 \\ 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{ll} 8 & 1 \\ 5 & 4 \end{array}\right] $$

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 \(A B\) and \(B A\) 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. $$\begin{array}{r}2 w+x-2 y-z=3 \\\w-2 x+y+z=4 \\\\-w-8 x+7 y+5 z=13 \\\3 w+x-2 y+2z=6\end{array}$$

In Exercises \(9-12,\) write the system of linear equations represented by the augmented matrix. Use \(x, y, z,\) and, if necessary, \(w, x, y,\) and \(z,\) for the variables. \(\left[\begin{array}{rrr|r}7 & 0 & 4 & -13 \\ 0 & 1 & -5 & 11 \\ 2 & 7 & 0 & 6\end{array}\right]\)