Chapter 8

Solve the system by the method of elimination. Label each line with its equation. $$\left\\{\begin{array}{c} 2 x+y=5 \\ x-y=1 \end{array}\right.$$

In general, when multiplying matrices \(A\) and \(B\), does \(A B=B A ?\)

Is a consistent system with exactly one solution independent or dependent?

Find the determinant of the matrix. $$\left[\begin{array}{rr} -5 & 2 \\ 6 & 3 \end{array}\right]$$

Use a determinant to find the area of the figure with the given vertices. \(\left(\frac{9}{2}, 0\right),(2,6),\left(0,-\frac{3}{2}\right)\).

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rr} -\frac{1}{2} & -\frac{5}{4} \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 8 & 5 \\ -4 & -2 \end{array}\right]$$

Determine the dimension of the matrix. $$\left[\begin{array}{llll} 3 & -1 & 2 & 6 \end{array}\right]$$

Determine whether each ordered pair is a solution of the system of equations. $$\left\\{\begin{array}{l} 4 x^{2}+y=3 \\ -x-y=11 \end{array}\right.$$ (a) (2,-13) (b) (-2,-9) (c) \(\left(-\frac{3}{2}, 6\right)\) (d) \(\left(-\frac{7}{4},-\frac{37}{4}\right)\)

Solve the system by the method of elimination. Label each line with its equation. $$\left\\{\begin{aligned} x+3 y &=1 \\ -x+2 y &=4 \end{aligned}\right.$$

What is the dimension of \(A B\) when \(A\) is a \(2 \times 3\) matrix and \(B\) is a \(3 \times 4\) matrix?

Is a consistent system with infinitely many solutions independent or dependent?

Find the determinant of the matrix. $$\left[\begin{array}{rr} -7 & 6 \\ \frac{1}{2} & 3 \end{array}\right]$$

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rrr} 2 & -17 & 11 \\ -1 & 11 & -7 \\ 0 & 3 & -2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 2 & 4 & -3 \\ 3 & 6 & -5 \end{array}\right]$$

Determine the dimension of the matrix. $$\left[\begin{array}{r} 4 \\ 32 \\ 3 \end{array}\right]$$

Determine whether each ordered pair is a solution of the system of equations. $$\left\\{\begin{aligned} y &=-2 e^{x} \\ 3 x-y &=2 \end{aligned}\right.$$ (a) (-2,0) (b) (0,-2) (c) (0,-3) (d) (-1,-5)

Solve the system by the method of elimination. Label each line with its equation. $$\left\\{\begin{aligned} x+y &=0 \\ 3 x+2 y &=1 \end{aligned}\right.$$

Equality of Matrices Find \(x\) and \(y\) or \(x, y,\) and \(z.\) $$\left[\begin{array}{lr} x & -7 \\ 9 & y \end{array}\right]=\left[\begin{array}{ll} 5 & -7 \\ 9 & -8 \end{array}\right]$$

Determine whether each ordered triple is a solution of the system of equations. \(\left\\{\begin{array}{rr}3 x-y+z= & 1 \\ 2 x-3 z= & -14 \\ 5 y+2 z= & 8\end{array}\right.\) (a) (3,5,-3) (b) (-1,0,4) (c) (0,-1,3) (d) (1,0,4)

Find the determinant of the matrix. $$\left[\begin{array}{lr}4 & -3 \\\0 & 0\end{array}\right]$$

The Inverse of a Matrix, show that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rrr} -4 & 1 & 5 \\ -1 & 2 & 4 \\ 0 & -1 & -1 \end{array}\right], \quad B=\frac{1}{4}\left[\begin{array}{rrr} -2 & 4 & 6 \\ 1 & -4 & -11 \\ -1 & 4 & 7 \end{array}\right]$$

Determine the dimension of the matrix. $$\left[\begin{array}{rrr} 5 & 4 & 2 \\ 3 & -5 & 1 \\ 7 & -2 & 9 \end{array}\right]$$

Determine whether each ordered pair is a solution of the system of equations. $$\left\\{\begin{aligned} -\log _{10} x+3 &=y \\ \frac{1}{9} x+y &=\frac{28}{9} \end{aligned}\right.$$ (a) (100,1) (b) (10,2) (c) (1,3) (d) (1,1)

Solve the system by the method of elimination. Label each line with its equation. $$\left\\{\begin{array}{l} 2 x-y=3 \\ 4 x+3 y=21 \end{array}\right.$$

Equality of Matrices Find \(x\) and \(y\) or \(x, y,\) and \(z.\) $$\left[\begin{array}{rr} -5 & x \\ y & 8 \end{array}\right]=\left[\begin{array}{rr} -5 & 13 \\ 12 & 8 \end{array}\right]$$

Determine whether each ordered triple is a solution of the system of equations. \(\left\\{\begin{array}{l}3 x+4 y-z=17 \\ 5 x-y+2 z=-2 \\ 2 x-3 y+7 z=-21\end{array}\right.\) (a) (1,5,6) (b) (-2,-4,2) (c) (1,3,-2) (d) (0,7,0)

Find the determinant of the matrix. $$\left[\begin{array}{rr}\sqrt{3} & 3 \\\4 & \sqrt{3}\end{array}\right]$$

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{ll} 2 & 0 \\ 0 & 3 \end{array}\right]$$

Determine the dimension of the matrix. $$\left[\begin{array}{rr} 33 & 45 \\ -9 & 20 \end{array}\right]$$

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{array}{c} 2 x+y=6 \\ -x+y=0 \end{array}\right.$$

Solve the system by the method of elimination. Label each line with its equation. $$\left\\{\begin{aligned} x-y &=2 \\ -2 x+2 y &=5 \end{aligned}\right.$$

Equality of Matrices Find \(x\) and \(y\) or \(x, y,\) and \(z.\) $$\left[\begin{array}{rrr} 4 & 5 & 4 \\ 13 & 15 & 3 y \\ 2 & 2 z-6 & 0 \end{array}\right]=\left[\begin{array}{rrr} 4 & 2 x+7 & 4 \\ 13 & 15 & 12 \\ 2 & 3 z-14 & 0 \end{array}\right]$$

Determine whether each ordered triple is a solution of the system of equations. \(\left\\{\begin{array}{rr}4 x+y-z= & 0 \\ -8 x-6 y+z= & -\frac{7}{4} \\ 3 x-y & =-\frac{9}{4}\end{array}\right.\) (a) (0,1,1) (b) \(\left(-\frac{3}{2}, \frac{5}{4},-\frac{5}{4}\right)\) (c) \(\left(-\frac{1}{2}, \frac{3}{4},-\frac{5}{4}\right)\) (d) \(\left(-\frac{1}{2}, \frac{1}{6},-\frac{3}{4}\right)\)

Find the determinant of the matrix. $$\left[\begin{array}{rr}9 & \sqrt{5} \\\\\sqrt{5} & 4\end{array}\right]$$

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{ll} 1 & 2 \\ 3 & 7 \end{array}\right]$$

Determine the dimension of the matrix. $$\left[\begin{array}{rrrr} 3 & -1 & 6 & 4 \\ -2 & 5 & 7 & 7 \end{array}\right]$$

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{array}{l} x-y=-4 \\ x+2 y=5 \end{array}\right.$$

Solve the system by the method of elimination. Label each line with its equation. $$\left\\{\begin{aligned} 3 x-2 y &=5 \\ -6 x+4 y &=-10 \end{aligned}\right.$$

Equality of Matrices Find \(x\) and \(y\) or \(x, y,\) and \(z.\) $$\left[\begin{array}{rrr} x+4 & 8 & -3 \\ 1 & 22 & 2 y \\ 7 & -2 & z+2 \end{array}\right]=\left[\begin{array}{rrr} 2 x+9 & 8 & -3 \\ 1 & 22 & -8 \\ 7 & -2 & 11 \end{array}\right]$$

Determine whether each ordered triple is a solution of the system of equations. \(\left\\{\begin{array}{rr}-4 x-y-8 z= & -6 \\ y+z= & 0 \\ 4 x-7 y & =6\end{array}\right.\) (a) (-2,-2,2) (b) \(\left(-\frac{33}{2},-10,10\right)\) (c) \(\left(\frac{1}{8},-\frac{1}{2}, \frac{1}{2}\right)\) (d) \(\left(-\frac{11}{2},-4,4\right)\)

Use the matrix capabilities of a graphing utility to find the determinant of the matrix. $$\left[\begin{array}{ll}1.9 & -0.3 \\\5.6 & 3.2\end{array}\right]$$

Use a determinant to determine whether the points are collinear. \((3,-1),(0,-3),(12,5)\)

Find the inverse of the matrix (if it exists). $$\left[\begin{array}{ll} 1 & -2 \\ 2 & -3 \end{array}\right]$$

Write the augmented matrix for the system of linear equations. What is the dimension of the augmented matrix? $$\left\\{\begin{aligned} 4 x-3 y &=-5 \\ -x+3 y &=12 \end{aligned}\right.$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} x+2 y=3 \\ x-2 y=1 \end{array}\right.$$

Solve the system by the method of substitution. Check your solution graphically. $$\left\\{\begin{array}{l} x-y=-4 \\ x^{2}-y=-2 \end{array}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 5 & -2 \\ 3 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & 1 \\ -2 & 6 \end{array}\right]$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} 2 x-y+5 z &=16 \\ y+2 z &=2 \\ z &=2 \end{aligned}\right.$$

Use a determinant to determine whether the points are collinear. \((3,-5),(6,1),(4,2)\)

Write the augmented matrix for the system of linear equations. What is the dimension of the augmented matrix? $$\left\\{\begin{array}{l} 7 x+4 y=22 \\ 5 x-9 y=15 \end{array}\right.$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} 3 x-5 y=2 \\ 2 x+5 y=13 \end{array}\right.$$