Chapter 8

Find the determinant of the matrix. Expand by cofactors on each indicated row or column. \(\left[\begin{array}{rrr}-3 & 4 & 2 \\ 6 & 3 & 1 \\ 4 & -7 & -8\end{array}\right]\) (a) Row 2 (b) Column 3

Use Cramer's Rule to solve (if possible) the system of equations. \(\left\\{\begin{array}{rr}6 x-5 y= & 17 \\ -13 x+3 y= & -76\end{array}\right.\)

Write the system of linear equations represented by the augmented matrix. (Use the variables \(x, y, z,\) and \(w,\) if applicable.) $$\left[\begin{array}{rrrrrr} 6 & 2 & -1 & -5 & 5 & -25 \\ -1 & 0 & 7 & 3 & \vdots & 7 \\ 4 & -1 & -10 & 6 & \vdots & 23 \\ 0 & 8 & 1 & -11 & \vdots & -21 \end{array}\right]$$

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} 1 & 2 & -1 \\ 3 & 7 & -10 \\ -5 & -7 & -15 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} 9 x+3 y=18 \\ 2 x-7 y=-19 \end{array}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{c} 6 x-3 y-4=0 \\ x+2 y-4=0 \end{array}\right.$$

Evaluating an Expression Evaluate the expression. $$\left[\begin{array}{rr} 6 & 9 \\ -1 & 0 \\ 7 & 1 \end{array}\right]+\left[\begin{array}{rr} 0 & 5 \\ -2 & -1 \\ 3 & -6 \end{array}\right]+\left[\begin{array}{rr} -13 & -7 \\ 4 & -1 \\ -6 & 0 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{aligned} x+y+z &=5 \\ x-2 y+4 z &=-1 \\ 3 y+4 z &=-1 \end{aligned}\right.$$

Find the determinant of the matrix. Expand by cofactors on each indicated row or column. \(\left[\begin{array}{rrrr}6 & 0 & -3 & 5 \\ 4 & 13 & 6 & -8 \\ -1 & 0 & 7 & 4 \\ 8 & 6 & 0 & 2\end{array}\right]\) (a) Row 2 (b) Column 2

Identify the elementary row operation performed to obtain the new row- equivalent matrix. New Row-Equivalent Matrix \(\left[\begin{array}{rrr}1 & -2 & 5 \\ -2 & 6 & 7\end{array}\right]\) \(\left[\begin{array}{rrr}-18 & 0 & 6 \\ 5 & 2 & -2\end{array}\right]\) \(\left[\begin{array}{rrrr}-1 & -2 & 3 & -2 \\ 2 & -5 & 1 & -7 \\ 0 & -6 & 8 & -4\end{array}\right]\) \(\left[\begin{array}{rrrr}-1 & 3 & -7 & 6 \\ 0 & -1 & -5 & 5 \\ 4 & -5 & 1 & 3\end{array}\right]\) Original Matrix \(\left[\begin{array}{ccc}-4 & 8 & -20 \\ -2 & 6 & 7\end{array}\right]\)

Use Cramer's Rule to solve (if possible) the system of equations. \(\left\\{\begin{array}{l}4 x-y+z=-5 \\ 2 x+2 y+3 z=10 \\ 6 x+y+4 z=-5\end{array}\right.\)

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} -\frac{1}{2} & \frac{3}{4} & \frac{1}{4} \\ 1 & 0 & -\frac{3}{2} \\ 0 & -1 & \frac{1}{2} \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{r} 1.8 x+1.2 y=4 \\ 9 x+6 y=3 \end{array}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{l} 1.5 x+0.8 y=2.3 \\ 0.3 x-0.2 y=0.1 \end{array}\right.$$

Evaluating an Expression Evaluate the expression. $$\frac{1}{3}\left(\left[\begin{array}{rrr} -4 & 0 & 1 \\ 0 & 2 & -12 \end{array}\right]-\left[\begin{array}{rrr} 5 & 1 & -2 \\ 12 & -6 & 3 \end{array}\right]\right)$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{aligned} 2 x\quad &+2 z=2 \\ 5 x+3 y\quad &=4 \\ \quad3 y-4 z &=4 \end{aligned}\right.$$

Find the determinant of the matrix. Expand by cofactors on each indicated row or column. \(\left[\begin{array}{rrrr}10 & 8 & 3 & -7 \\ 4 & 0 & 5 & -6 \\ 0 & 3 & 2 & 7 \\\ 1 & 0 & -3 & 2\end{array}\right]\) (a) Row 3 (b) Column I

Use Cramer's Rule to solve (if possible) the system of equations. \(\left\\{\begin{array}{l}4 x-2 y+3 z=-2 \\ 2 x+2 y+5 z=16 \\ 8 x-5 y-2 z=4\end{array}\right.\)

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} -\frac{5}{6} & \frac{1}{3} & \frac{11}{6} \\ 0 & \frac{2}{3} & 2 \\ 1 & -\frac{1}{2} & -\frac{5}{2} \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} 3.1 x-2.9 y=-10.2 \\ 31 x-12 y=34 \end{array}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} -0.5 x+4 y &=7.8 \\ 0.2 x-1.6 y &=-3.6 \end{aligned}\right.$$

Evaluating an Expression Evaluate the expression. $$\frac{1}{2}\left(\left[\begin{array}{ccc} 3 & -2 & 4 & 0 \end{array}\right]-\left[\begin{array}{cccc} 10 & -6 & -18 & 9 \end{array}\right]\right)$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{r} 2 x+4 y+z=2 \\ -2 y-3 z=-8 \\ x-z=-1 \end{array}\right.$$

Solve the system of equations using (a) Gaussian elimination and (b) Cramer's Rule. Which method do you prefer, and why? \(\left\\{\begin{array}{l}3 x+3 y+5 z=1 \\ 3 x+5 y+9 z=2 \\ 5 x+9 y+17 z=4\end{array}\right.\)

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} 0.1 & 0.2 & 0.3 \\ -0.3 & 0.2 & 0.2 \\ 0.5 & 0.4 & 0.4 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} 3 x+\frac{1}{4} y=1 \\ 2 x-\frac{1}{3} y=0 \end{array}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{c} \frac{1}{5} x+\frac{1}{2} y=8 \\ x+y=20 \end{array}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to the nearest thousandths, if necessary. $$\frac{3}{7}\left[\begin{array}{rr} 2 & 5 \\ -1 & -4 \end{array}\right]+6\left[\begin{array}{rr} -3 & 0 \\ 2 & 2 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{cc} 4 x+y-3 z= & 11 \\ 2 x-3 y+2 z= & 9 \\ x+y+z= & -3 \end{array}\right.$$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{rrr}-3 & 1 & 0 \\\7 & 11 & 5 \\\1 & 2 & 2\end{array}\right]$$

Solve the system of equations using (a) Gaussian elimination and (b) Cramer's Rule. Which method do you prefer, and why? \(\left\\{\begin{array}{l}2 x+3 y-5 z=1 \\ 3 x+5 y+9 z=-16 \\ 5 x+9 y+17 z=-30\end{array}\right.\)

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrr} 0.6 & 0 & -0.3 \\ 0.7 & -1 & 0.2 \\ 1 & 0 & -0.9 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{c} \frac{1}{2} x-2 y=-\frac{5}{2} \\ -x+4 y=5 \end{array}\right.$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{l} \frac{1}{2} x+\frac{3}{4} y=10 \\ \frac{3}{4} x-y=4 \end{array}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to the nearest thousandths, if necessary. $$\frac{4}{5}\left[\begin{array}{rr} 14 & -11 \\ -22 & 19 \end{array}\right]+7\left[\begin{array}{rr} -22 & 20 \\ 13 & 6 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{c} 5 x-3 y+2 z=3 \\ 2 x+4 y-z=7 \\ x-11 y+4 z=3 \end{array}\right.$$

Fill in the blank(s) using elementary row operations to form a row-equivalent matrix. $$\begin{aligned} &\left[\begin{array}{ccc} 1 & 4 & 3 \\ 2 & 10 & 5 \end{array}\right]\\\ &\left[\begin{array}{rrr} 1 & 4 & 3 \\ 0 & & -1 \end{array}\right] \end{aligned}$$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{rrr} 6 & 3 & -7 \\ 0 & 0 & 0 \\ 4 & -6 & 3 \end{array}\right]$$

The retail sales of family clothing stores in the United States from 2009 through 2013 are shown in the table. The coefficients of the least squares regression parabola \(y=a t^{2}+b t+c,\) where \(y\) represents the retail sales (in billions of dollars) and \(t\) represents the year, with \(t=9\) corresponding to \(2009,\) can be found by solving the system \(\left\\{\begin{array}{rr}80,499 a+6985 b+615 c= & 56,453.6 \\\ 6985 a+615 b+55 c= & 5004.4 \\ 615 a+\quad 55 b+\quad 5 c= & 450.8\end{array}\right.\) (a) Use Cramer's Rule to solve the system and write the least squares regression parabola for the data. (b) Use a graphing utility to graph the parabola with the data. How well does the model fit the data? (c) Use the model to predict the retail sales of family clothing stores in the U.S. in the year 2015

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrrr} -1 & 0 & 1 & 0 \\ 0 & 2 & 0 & -2 \\ 2 & 0 & -1 & 0 \\ 0 & -1 & 0 & 1 \end{array}\right]$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{array}{l} -\frac{5}{3} x+y=5 \\ -5 x+3 y=6 \end{array}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to the nearest thousandths, if necessary. $$-5\left[\begin{array}{rr} 3.211 & 6.829 \\ -1.004 & 4.914 \\ 0.055 & -3.889 \end{array}\right]-\frac{1}{4}\left[\begin{array}{rr} 1.630 & -3.090 \\ 5.256 & 8.335 \\ -9.768 & 4.251 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{aligned} 3 x-2 y+4 z &=1 \\ x+y-2 z &=3 \\ 2 x-3 y+6 z &=8 \end{aligned}\right.$$

The retail sales \(y\) (in billions of dollars) of stores selling auto parts, accessories, and tires in the United States from 2009 through 2013 are given by the ordered pairs of the form \((t, y(t)),\) where \(t=9\) represents \(2009 .\) $$\begin{aligned} &(9,74.1) \quad(10,77.7) \quad(11,82.7)\\\ &(12,83.9) \quad(13,82.8) \end{aligned}$$ The coefficients of the least squares regression parabola \(y=a t^{2}+b t+c\) can be found by solving the system \(\left\\{\begin{array}{rr}80,499 a+6985 b+615 c= & 49,853.8 \\ 6985 a+615 b+55 c= & 4436.8 \\ 615 a+\quad 55 b+\quad 5 c= & 401.2\end{array}\right.\) (a) Use Cramer's Rule to solve the system and write the least squares regression parabola for the data. (b) Use a graphing utility to graph the parabola with the data. How well does the model fit the data? (c) Is this a good model for predicting retail sales in future years? Explain.

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{rrr}1 & 1 & 2 \\\3 & -5 & 9 \\\0 & 0 & 0\end{array}\right]$$

Finding the Inverse of a Matrix, use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$\left[\begin{array}{rrrr} 1 & -2 & -1 & -2 \\ 3 & -5 & -2 & -3 \\ 2 & -5 & -2 & -5 \\ -1 & 4 & 4 & 11 \end{array}\right]$$

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} -\frac{2}{3} x+y &=2 \\ 3 x-\frac{1}{2} y &=4 \end{aligned}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to the nearest thousandths, if necessary. $$-3\left[\begin{array}{rr} 10 & 15 \\ -20 & 10 \\ 12 & 4 \end{array}\right]-\frac{1}{8}\left(\left[\begin{array}{rr} 12 & 11 \\ 7 & 0 \\ 6 & 9 \end{array}\right]+\left[\begin{array}{rr} -3 & 13 \\ -3 & 8 \\ -14 & 15 \end{array}\right]\right)$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{l} 2 x+4 y+z=-4 \\ 2 x-4 y+6 z=13 \\ 4 x-2 y+z=6 \end{array}\right.$$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{rrr}-1 & 2 & -5 \\\0 & 3 & -4 \\\0 & 0 & 3\end{array}\right]$$