Chapter 5

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{rr|r}1 & 0 & 3 \\ 0 & 1 & -2\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} 4 x-5 y &=14 \\ 2 x+3 y &=-4 \end{aligned}\)

Compute the indicated products. \(\left[\begin{array}{rrr}3 & 2 & -1 \\ 4 & -1 & 0 \\ -5 & 2 & 1\end{array}\right]\left[\begin{array}{r}3 \\ -2 \\ 0\end{array}\right]\)

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) Compute \(2 A-3 B\).

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrrr|r}0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 1 & -2 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{ll|l}1 & 1 & 3 \\ 0 & 0 & 0\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\frac{5}{4} x-\frac{2}{3} y=3\) \(\frac{1}{4} x+\frac{5}{3} y=6\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrr}4 & 2 & 2 \\ -1 & -3 & 4 \\ 3 & -1 & 6\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{rr}-1 & 2 \\ 3 & 1\end{array}\right]\left[\begin{array}{ll}2 & 4 \\ 3 & 1\end{array}\right]\)

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) Compute \(C-D\).

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrrr|r}1 & 0 & 3 & 0 & 2 \\ 0 & 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{ll|l}0 & 1 & 3 \\ 1 & 0 & 5\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} 2 x-3 y &=6 \\ 6 x-9 y &=12 \end{aligned}\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrr}1 & 2 & 0 \\ -3 & 4 & -2 \\ -5 & 0 & -2\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{rr}1 & 3 \\ -1 & 2\end{array}\right]\left[\begin{array}{lll}1 & 3 & 0 \\ 3 & 0 & 2\end{array}\right]\)

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) Compute \(4 D-2 C\).

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrrr|r}1 & 0 & 3 & -1 & 4 \\ 0 & 1 & -2 & 3 & 2 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{ll|l}0 & 1 & 3 \\ 0 & 0 & 5\end{array}\right]\)

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} \frac{2}{3} x+y &=5 \\ \frac{1}{2} x+\frac{3}{4} y &=\frac{15}{4} \end{aligned}\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrr}1 & 4 & -1 \\ 2 & 3 & -2 \\ -1 & 2 & 3\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{lll}2 & 1 & 2 \\ 3 & 2 & 4\end{array}\right]\left[\begin{array}{rr}-1 & 2 \\ 4 & 3 \\ 0 & 1\end{array}\right]\)

Perform the indicated operations. \(\left[\begin{array}{lll}6 & 3 & 8 \\ 4 & 5 & 6\end{array}\right]-\left[\begin{array}{lll}3 & -2 & -1 \\ 0 & -5 & -7\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} 2 x-y &=3 \\ x+2 y &=4 \\ 2 x+3 y &=7 \end{aligned}\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{lll|l}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 5\end{array}\right]\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrr}3 & -2 & 7 \\ -2 & 1 & 4 \\ 6 & -5 & 8\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{rr}-1 & 2 \\ 4 & 3 \\ 0 & 1\end{array}\right]\left[\begin{array}{lll}2 & 1 & 2 \\ 3 & 2 & 4\end{array}\right]\)

Perform the indicated operations. \(\left[\begin{array}{rrrr}2 & -3 & 4 & -1 \\ 3 & 1 & 0 & 0\end{array}\right]+\left[\begin{array}{rrrr}4 & 3 & -2 & -4 \\ 6 & 2 & 0 & -3\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} x+2 y &=3 \\ 2 x-3 y &=-8 \\ x-4 y &=-9 \end{aligned}\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{lll|l}1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 2 & -3\end{array}\right]\)

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrrr}1 & 1 & -1 & 1 \\ 2 & 1 & 1 & 0 \\ 2 & 1 & 0 & 1 \\\ 2 & -1 & -1 & 3\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{ll}0.1 & 0.9 \\ 0.2 & 0.8\end{array}\right]\left[\begin{array}{ll}1.2 & 0.4 \\ 0.5 & 2.1\end{array}\right]\)

Perform the indicated operations. \(\left[\begin{array}{rrr}1 & 4 & -5 \\ 3 & -8 & 6\end{array}\right]+\left[\begin{array}{rrr}4 & 0 & -2 \\ 3 & 6 & 5\end{array}\right]-\left[\begin{array}{rrr}2 & 8 & 9 \\ -11 & 2 & -5\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} 3 x-2 y &=-3 \\ 2 x+y &=3 \\ x-2 y &=-5 \end{aligned}\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{rrr|r}1 & 0 & 1 & 3 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & -1 & 6\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. AGRICULTURE The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost of cultivating corn and wheat (including seeds and labor) is $$\$ 42$$ and $$\$ 30$$ per acre, respectively. Jacob Johnson has $$\$ 18,600$$ available for cultivating these crops. If he wishes to use all the allotted land and his entire budget for cultivating these two crops, how many acres of each crop should he plant?

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrrr}1 & 1 & 2 & 3 \\ 2 & 3 & 0 & -1 \\ 0 & 2 & -1 & 1 \\\ 1 & 2 & 1 & 1\end{array}\right]\)

Compute the indicated products. \(\left[\begin{array}{ll}1.2 & 0.3 \\ 0.4 & 0.5\end{array}\right]\left[\begin{array}{rr}0.2 & 0.6 \\ 0.4 & -0.5\end{array}\right]\)

Perform the indicated operations. \(3\left[\begin{array}{rrr}1 & 1 & -3 \\ 3 & 2 & 3 \\ 7 & -1 & 6\end{array}\right]+4\left[\begin{array}{rrr}-2 & -1 & 8 \\ 4 & 2 & 2 \\ 3 & 6 & 3\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}2 x+3 y= & 2 \\ x+3 y= & -2 \\ x-y= & 3\end{array}\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{rr|r}1 & 0 & -10 \\ 0 & 1 & 2 \\ 0 & 0 & 0\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. Michael Perez has a total of $$\$ 2000$$ on deposit with two savings institutions. One pays interest at the rate of $$6 \% /$$ year, whereas the other pays interest at the rate of $$8 \% /$$ year. If Michael earned a total of $$\$ 144$$ in interest during a single year, how much does he have on deposit in each institution?

Compute the indicated products. \(\left[\begin{array}{rrr}6 & -3 & 0 \\ -2 & 1 & -8 \\ 4 & -4 & 9\end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\)

Perform the indicated operations. \(\left[\begin{array}{lll}1.2 & 4.5 & -4.2 \\ 8.2 & 6.3 & -3.2\end{array}\right]-\left[\begin{array}{rrr}3.1 & 1.5 & -3.6 \\ 2.2 & -3.3 & -4.4\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} 3 x-2 y &=5 \\\\-x+3 y &=-4 \\ 2 x-4 y &=6 \end{aligned}\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{lll|l}0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. The Coffee Shoppe sells a coffee blend made from two coffees, one costing $$\$ 5 / \mathrm{lb}$$ and the other costing $$\$ 6 / \mathrm{lb}$$. If the blended coffee sells for $$\$ 5.60 / \mathrm{lb}$$, find how much of each coffee is used to obtain the desired blend. Assume that the weight of the blended coffee is \(100 \mathrm{lb}\).

Compute the indicated products. \(\left[\begin{array}{rr}2 & 4 \\ -1 & -5 \\ 3 & -1\end{array}\right]\left[\begin{array}{rrr}2 & -2 & 4 \\ 1 & 3 & -1\end{array}\right]\)

Perform the indicated operations. \(\left[\begin{array}{rr}0.06 & 0.12 \\ 0.43 & 1.11 \\ 1.55 & -0.43\end{array}\right]-\left[\begin{array}{ll}0.77 & -0.75 \\ 0.22 & -0.65 \\\ 1.09 & -0.57\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} 4 x+6 y &=8 \\ 3 x-2 y &=-7 \\ x+3 y &=5 \end{aligned}\)

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{lll|l}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 4 \\\ 0 & 0 & 1 & 5\end{array}\right]\)