Chapter 8

You are given the yearly interest earned from a total of \$18,000 invested in two funds paying the given rates of simple interest. Write and solve a system of equations to find the amount invested at each rate. Yearly Interest \(\$ 400\) Rate 1 \(4 \%\) Rate 2 \(2 \%\)

Solving a Matrix Equation Solve for \(X\) when \(A=\left[\begin{array}{rr}-2 & -1 \\\ 1 & 0 \\ 3 & -4\end{array}\right]\) and \(B=\left[\begin{array}{rr}0 & 3 \\\ 2 & 0 \\ -4 & -1\end{array}\right]\) $$3 X=A+3 B$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{l} 3 x+3 y+5 z=1 \\ 3 x+5 y+9 z=0 \\ 5 x+9 y+17 z=0 \end{array}\right.$$

Fill in the blank(s) using elementary row operations to form a row-equivalent matrix. $$\begin{aligned} &\left[\begin{array}{rrrr} 2 & 4 & 8 & 3 \\ 1 & -1 & -3 & 2 \\ 2 & 6 & 4 & 9 \end{array}\right]\\\ &\left[\begin{array}{cccc} 1 & -1 & -3 & 2 \\ 2 & & & \\ 2 & 6 & 4 & 9 \end{array}\right]\\\ &\left[\begin{array}{cccc} 1 & -1 & -3 & 2 \\ 0 & 6 & & \\ 0 & 8 & & \end{array}\right] \end{aligned}$$

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

You are given the yearly interest earned from a total of \$18,000 invested in two funds paying the given rates of simple interest. Write and solve a system of equations to find the amount invested at each rate. Yearly Interest \$ 840 Rate 1 \(6 \%\) Rate 2 \(3 \%\)

Solving a Matrix Equation Solve for \(X\) when \(A=\left[\begin{array}{rr}-2 & -1 \\\ 1 & 0 \\ 3 & -4\end{array}\right]\) and \(B=\left[\begin{array}{rr}0 & 3 \\\ 2 & 0 \\ -4 & -1\end{array}\right]\) $$2 X=2 A+B$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{l} 2 x+y+3 z=1 \\ 2 x+6 y+8 z=3 \\ 6 x+8 y+18 z=5 \end{array}\right.$$

(a) perform the row operations to solve the augmented matrix, (b) write and solve the system of linear equations represented by the augmented matrix, and (c) compare the two solution methods. Which do you prefer? $$\left[\begin{array}{rrrr} -3 & 4 & \vdots & 22 \\ 6 & -4 & \vdots & -28 \end{array}\right]$$ (i) Add \(R_{2}\) to \(R_{1}\) (ii) Add -2 times \(R_{1}\) to \(R_{2}\) (iii) Multiply \(R_{2}\) by \(-\frac{1}{4}\) (iv) Multiply \(R_{1}\) by \(\frac{1}{3}\)

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{llll} 2 & 6 & 6 & 2 \\ 2 & 7 & 3 & 6 \\ 1 & 5 & 0 & 1 \\ 3 & 7 & 0 & 7 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}5 x+3 y=6 \\ 3 x-y=5\end{array}\right.\)

You are given the yearly interest earned from a total of \$18,000 invested in two funds paying the given rates of simple interest. Write and solve a system of equations to find the amount invested at each rate. Yearly Interest \(\$ 1182\) Rate 1 \(5.6 \%\) Rate 2 \(6.8 \%\)

Solving a Matrix Equation Solve for \(X\) when \(A=\left[\begin{array}{rr}-2 & -1 \\\ 1 & 0 \\ 3 & -4\end{array}\right]\) and \(B=\left[\begin{array}{rr}0 & 3 \\\ 2 & 0 \\ -4 & -1\end{array}\right]\) $$2 A+4 B=-2 X$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{aligned} 3 x-3 y+6 z &=6 \\ x+2 y-z &=5 \\ 5 x-8 y+13 z &=7 \end{aligned}\right.$$

(a) perform the row operations to solve the augmented matrix, (b) write and solve the system of linear equations represented by the augmented matrix, and (c) compare the two solution methods. Which do you prefer? $$\left[\begin{array}{rrrrr} 7 & 13 & 1 & \vdots & -4 \\ -3 & -5 & -1 & \vdots & -4 \\ 3 & 6 & 1 & \vdots & -2 \end{array}\right]$$ (i) Add \(R_{2}\) to \(R_{1}\) (ii) Multiply \(R_{1}\) by \(\frac{1}{4}\) (iii) Add \(R_{3}\) to \(R_{2}\) (iv) Add -3 times \(R_{1}\) to \(R_{3}\) (v) Add -2 times \(R_{2}\) to \(R_{1}\)

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{rrrr} 3 & 6 & -5 & 4 \\ -2 & 2 & 6 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 3 & -1 & -1 \end{array}\right]$$

Finding the Inverse of a \(2 \times 2\) Matrix, use the formula on page 676 to find the inverse of the \(2 \times 2\) matrix (if it exists). $$\left[\begin{array}{rr} -\frac{1}{4} & -\frac{2}{3} \\ \frac{1}{3} & \frac{8}{9} \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{c}x+5 y=10 \\ 3 x-10 y=-5\end{array}\right.\)

You are given the yearly interest earned from a total of \$18,000 invested in two funds paying the given rates of simple interest. Write and solve a system of equations to find the amount invested at each rate. Yearly Interest \(\$ 684\) Rate 1 \(2.75 \%\) Rate 2 \(4.25 \%\)

Solving a Matrix Equation Solve for \(X\) when \(A=\left[\begin{array}{rr}-2 & -1 \\\ 1 & 0 \\ 3 & -4\end{array}\right]\) and \(B=\left[\begin{array}{rr}0 & 3 \\\ 2 & 0 \\ -4 & -1\end{array}\right]\) $$-3 X-3 A=9 B$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{rr} -x+3 y+z= & 4 \\ 4 x-2 y-5 z= & -7 \\ 2 x+4 y-3 z= & 12 \end{array}\right.$$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{rrrrr} 3 & 2 & 4 & -1 & 5 \\ -2 & 0 & 1 & 3 & 2 \\ 1 & 0 & 0 & 4 & 0 \\ 6 & 0 & 2 & -1 & 0 \\ 3 & 0 & 5 & 1 & 0 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}\frac{2}{5} x-\frac{3}{2} y=4 \\ \frac{1}{5} x-\frac{3}{4} y=-2\end{array}\right.\)

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} x^{2}-2 x+y &=8 \\ x-y &=-2 \end{aligned}\right.$$

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{rr} 3 & -1 \\ 4 & -5 \\ 2 & 6 \end{array}\right], \quad B=\left[\begin{array}{rr} 6 & 0 \\ 7 & -1 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{rr} x-2 y+3 z= & 4 \\ 3 x-y+2 z= & 0 \\ x+3 y-4 z= & -2 \end{array}\right.$$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{lllll} 5 & 2 & 0 & 0 & -2 \\ 0 & 1 & 4 & 3 & 2 \\ 0 & 0 & 2 & 6 & 3 \\ 0 & 0 & 3 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2 \end{array}\right]$$

Finding the Inverse of a \(2 \times 2\) Matrix, use the formula on page 676 to find the inverse of the \(2 \times 2\) matrix (if it exists). $$\left[\begin{array}{rr} 7 & 12 \\ -8 & -5 \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}\frac{2}{3} x+\frac{1}{6} y=\frac{2}{3} \\ 4 x+y=4\end{array}\right.\)

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} 2 x^{2}-2 x-y &=14 \\ 2 x-y &=-2 \end{aligned}\right.$$

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{rr} -1 & 6 \\ -4 & 5 \\ 0 & 3 \end{array}\right], \quad B=\left[\begin{array}{ll} 2 & 3 \\ 0 & 9 \end{array}\right]$$

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{l} x\quad=13 \\ 4 x-2 y+z=1 \\ 2 x-2 y-7 z=-19 \end{array}\right.$$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$\left|\begin{array}{cccc} 1 & -1 & 8 & 4 \\ 2 & 6 & 0 & -4 \\ 2 & 0 & 2 & 6 \\ 0 & 2 & 8 & 0 \end{array}\right|$$

Finding a Matrix Entry, find the value of the constant \(k\) such that \(B=A^{-1}\). $$A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 0 \end{array}\right], \quad B=\left[\begin{array}{rr} k & -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{4} \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}\frac{3}{4} x+y=\frac{1}{8} \\ \frac{9}{4} x+3 y=\frac{3}{8}\end{array}\right.\)

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

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

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{llll} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 \end{array}\right]$$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$\left|\begin{array}{rrrr} 0 & -3 & 8 & 2 \\ 8 & 1 & -1 & 6 \\ -4 & 6 & 0 & 9 \\ -7 & 0 & 0 & 14 \end{array}\right|$$

Finding a Matrix Entry, find the value of the constant \(k\) such that \(B=A^{-1}\). $$A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} -\frac{1}{3} & \frac{1}{3} \\ k & \frac{1}{3} \end{array}\right]$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{aligned} \frac{1}{4} x+\frac{1}{6} y &=1 \\\\-3 x-2 y &=0 \end{aligned}\right.\)

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

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{r} 3 x-2 y-6 z=-4 \\ -3 x+2 y+6 z=1 \\ x-y-5 z=-3 \end{array}\right.$$

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. $$\left[\begin{array}{rrrr} 3 & 0 & 3 & 7 \\ 0 & -2 & 0 & 4 \\ 0 & 0 & 1 & 5 \end{array}\right]$$

Determine whether the statement is true or false. Justify your answer. Cramer's Rule cannot be used to solve a system of linear equations when the determinant of the coefficient matrix is zero.

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$\left|\begin{array}{rrrrrr} 3 & -2 & 4 & 3 & 1 & 3 \\ -1 & 0 & 2 & 1 & 0 & 0 \\ 5 & -1 & 0 & 3 & 2 & 1 \\ 4 & 7 & -8 & 0 & 0 & -2 \\ 1 & 2 & 3 & 0 & 2 & 4 \\ 3 & -5 & 1 & 2 & 3 & 1 \end{array}\right|$$

Solve the system by the method of elimination and check any solutions using a graphing utility. $$\left\\{\begin{array}{r} \frac{x+2}{4}+\frac{y-1}{4}=1 \\ x-y=4 \end{array}\right.$$

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

Finding the Product of Two Matrices Find \(A B,\) if possible. $$A=\left[\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -2 \end{array}\right], \quad B=\left[\begin{array}{rrr} \frac{1}{3} & 0 & 0 \\ 0 & -\frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{6} \end{array}\right]$$