Chapter 6

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

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. $$\left\\{\begin{array}{c}5 x+12 y+z=10 \\\2 x+5 y+2 z=1 \\\x+2 y-3 z=5\end{array}\right.$$

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

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

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

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. $$\left\\{\begin{array}{c}2 x-4 y+z=3 \\\x-3 y+z=5 \\\3 x-7 y+2 z=12\end{array}\right.$$

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

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

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

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

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

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

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. $$\left[\begin{array}{rrrr}-4 & 1 & 3 & -5 \\\2 & -1 & \pi & 0 \\\1 & 0 & -e & 5\end{array}\right]$$

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

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

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

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. $$\left\\{\begin{array}{c}3 x+4 y+2 z=3 \\\4 x-2 y-8 z=-4 \\\x+y-z=3\end{array}\right.$$

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

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

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

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

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

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

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. $$\left|\begin{array}{ll}-5 & -1 \\\\-2 & -7\end{array}\right|$$

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

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

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

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

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

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

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

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

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

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. $$ \left[\begin{array}{rrr|r} 5 & 0 & 3 & -11 \\ 0 & 1 & -4 & 12 \\ 7 & 2 & 0 & 3 \end{array}\right] $$

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

Find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative imerse of \(A\). $$A=\left[\begin{array}{lll}0 & 2 & 0 \\\3 & 3 & 2 \\\2 & 5 & 1\end{array}\right], \quad B=\left[\begin{array}{rrr}-3.5 & -1 & 2 \\\0.5 & 0 & 0 \\\45 & 2 & -3\end{array}\right]$$

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

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

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. $$ \left[\begin{array}{llr|r} 7 & 0 & 4 & -13 \\ 0 & 1 & -5 & 11 \\ 2 & 7 & 0 & 6 \end{array}\right] $$

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

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

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$\left\\{\begin{aligned}2 w+x-y \quad\quad &=3 \\\w-3 x+2 y\quad\quad &=-4 \\\3 w+x-3 y+z &=1 \\\w+2 x-4 y-z &=-2 \end{aligned}\right.$$

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. $$ \left[\begin{array}{rrrr|r} 1 & 1 & 4 & 1 & 3 \\ -1 & 1 & -1 & 0 & 7 \\ 2 & 0 & 0 & 5 & 11 \\ 0 & 0 & 12 & 4 & 5 \end{array}\right] $$

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$A=\left[\begin{array}{rrr}3 & 1 & 1 \\\\-1 & 2 & 5\end{array}\right], \quad B=\left[\begin{array}{rrr}2 & -3 & 6 \\\\-3 & 1 & -4\end{array}\right]$$