Write the augmented matrix corresponding to each system of equations. $$ \begin{array}{l} 2 x-3 y=7 \\ 3 x+y=4 \end{array} $$

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{r} 3 x+y=1 \\ -7 x-2 y=-1 \end{array} $$

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{r} x+y+z=0 \\ 2 x-y+z=1 \\ x+y-2 z=2 \end{array} $$

Matrix \(A\) is an input-output matrix associated with an economy, and matrix \(D\) (units in millions of dollars) is a demand vector. In each problem,find the final outputs of each industry such that the demands of industry and the consumer sector are met. $$ A=\left[\begin{array}{lll} \frac{1}{5} & \frac{2}{5} & \frac{1}{5} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & \frac{1}{5} & 0 \end{array}\right] \text { and } D=\left[\begin{array}{r} 10 \\ 5 \\ 15 \end{array}\right] $$

A dietitian plans a meal around three foods. The number of units of vitamin A, vitamin \(\mathrm{C}\), and calcium in each ounce of these foods is represented by the matrix \(M\), where $$ \begin{array}{l} \text { Food I } & \text { Food II } & \text { Food III } \\ \text { Vitamin A } & {\left[\begin{array}{rrr} 400 & 1200 & 800 \\ M= & \text { Vitamin C } \\ \text { Calcium } \end{array}\right.} & \begin{array}{rr} 110 \\ 90 \end{array} & \begin{array}{r} 570 \\ 30 \end{array} & \left.\begin{array}{r} 340 \\ 60 \end{array}\right] \end{array} $$ The matrices \(A\) and \(B\) represent the amount of each food (in ounces) consumed by a girl at two different meals, where $\begin{array}{lll}\text { Food I } & \text { Food II } & \text { Food III }\end{array}$ $$ A=\left[\begin{array}{lll} 7 & 1 & 6 \end{array}\right] $$ $\begin{array}{lll}\text { Food I } & \text { Food II } & \text { Food III }\end{array}$ $$ B=\left[9 \quad \left[\begin{array}{ll} 9 & 3 \end{array}\right.\right. $$ $$ 2] $$ Calculate the following matrices and explain the meaning of the entries in each matrix. a. \(M A^{T}\) b. \(M B^{T}\) c. \(M(A+B)^{T}\)

A university admissions committee anticipates an enrollment of 8000 students in its freshman class next year. To satisfy admission quotas, incoming students have been categorized according to their sex and place of residence. The number of students in each category is given by the matrix $$ \begin{array}{l} \text { In-state } \\ \text { A= Out-of-state } \\ \text { Foreign } \end{array}\left[\begin{array}{rr} 2700 & 3000 \\ 800 & 700 \\ 500 & 300 \end{array}\right] $$ By using data accumulated in previous years, the admissions committee has determined that these students will elect to enter the College of Letters and Science, the College of Fine Arts, the School of Business Administration, and the School of Engineering according to the percentages that appear in the following matrix: $$ B=\begin{array}{l} \text { Male } \\ \text { Female } \end{array}\left[\begin{array}{llll} 0.25 & 0.20 & 0.30 & 0.25 \\ 0.30 & 0.35 & 0.25 & 0.10 \end{array}\right] $$ Find the matrix \(A B\) that shows the number of in-state, outof-state, and foreign students expected to enter each discipline.