Exam 7: Systems of Equations and Matrices

Solve using Cramer's Rule. -Twice the water flow in the hot-water pipe is the same as three times the flow in the cold-water pipe. The combined flow is 1400 liters per hour. What is the flow in each pipe?

Find the indicated sum or difference, if it is defined. -Compute the difference $\mathrm { A } - \mathrm { B }$ if $\mathrm { A } = \left[ \begin{array} { r r } - 1 & 0 \\ 6 & 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 1 & 6 \\ 3 & 1 \end{array} \right]$ .

Find the solution or solutions, if any exist, to the system. - $\left\{ \begin{array} { l } x + y + z = 7 \\x - y + 2 z = 7\end{array} \right.$

Find the inverse matrix of A. - $A = \left[ \begin{array} { r r r r } 3 & - 2 & 0 & 2 \\9 & - 4 & 0 & 4 \\6 & - 8 & 0 & 6 \\0 & 0 & - 3 & 0\end{array} \right]$

Use an inverse matrix to find the solution to the system. - $\left\{ \begin{array} { c } 3 x - y - 5 z = - 34 \\- 3 x + 4 y + 3 z = 37 \\9 x - 6 y + z = 21\end{array} \right.$

Solve the system graphically. -A picture frame has a perimeter of $52 \mathrm { in }$ . and a diagonal of $\sqrt { 410 } \mathrm { in }$ . What are the dimensions?

Solve the matrix equation. - $\left[ \begin{array} { r r r } 2 & 8 & 6 \\ 4 & 2 & - 2 \\ 3 & - 1 & 1 \end{array} \right] \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 20 \\ - 2 \\ 11 \end{array} \right]$

Find the indicated matrix. -Let $B = \left[ \begin{array} { l l l l } - 1 & 6 & 4 & - 3 \end{array} \right]$ . Find -2B.

Solve the system graphically. -A rectangular plot has area 126 $\mathrm { yd } ^ { 2 }$ with a perimeter of 46 yd. What is the length of the longest side?

Find the value of the determinant. - $\left| \begin{array} { l l } - 7 & - 3 \\- 2 & 5\end{array} \right|$

The following system does not have a unique solution. Solve the system. - $\left\{ \begin{aligned} x + 5 y + 2 z & = - 5 \\ - 3 x + 5 y - 6 z & = 35 \end{aligned} \right.$

Use an inverse matrix to find the solution to the system. -Two phone companies compete for customers in a city. Company 1 retains $3 / 4$ of its customers and loses $1 / 4$ of its customers to Company 2 . Company 2 retains $5 / 6$ of its customers and loses $1 / 6$ to Company 1 . If we represent the fraction of the market held last year by $\left[ \begin{array} { l } a \\ b \end{array} \right]$ , where $a$ is the number the that Company 1 had last year and $b$ is number that Company 2 had last year, then the number that each company will have this year can be found by t] following matrix equation. $\left[ \begin{array} { l } \mathrm { A } \\\mathrm { B }\end{array} \right] = \left[ \begin{array} { l l } 3 / 4 & 1 / 6 \\1 / 4 & 5 / 6\end{array} \right] \left[ \begin{array} { l } \mathrm { a } \\\mathrm { b }\end{array} \right]$ If Company 1 has $\mathrm { A } = 90,000$ customers and Company 2 has $\mathrm { B } = 198,000$ customers this year, how many customers did Company 1 have last year?

Compute AB, if possible. - $A = \left[ \begin{array} { r r r } 1 & 3 & - 2 \\ 4 & 0 & 5 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 1 \\ 0 & 5 \end{array} \right]$

A matrix has an inverse if and only if its determinant does not equal 0. Determine whether the given matrix has an inverse. - $\left[ \begin{array} { l l l } 3 & 6 & 5 \\6 & 4 & 6 \\2 & 3 & 5\end{array} \right]$

first two games of the season. Write a matrix containing the total number of points and rebounds for each of the starting five. Game 1 Points Rebounds Levy 20 3 Cowens 16 5 Williams 8 12 Miller 6 11 Jenkins 10 2 Game 2 Points Rebounds Levy 18 4 Cowens 14 3 Williams 12 9 Miller 4 10 Jenkins 10 3 A) $\left[ \begin{array} { l l } 5 & 62 \end{array} \right]$ В) $\left[ \begin{array} { r r } 7 & 38 \\ 30 & 8 \\ 21 & 20 \\ 21 & 10 \\ 5 & 20 \end{array} \right]$ C) $\left[ \begin{array} { r r } 38 & 7 \\ 30 & 8 \\ 20 & 21 \\ 10 & 21 \\ 20 & 5 \end{array} \right]$ D) $[ 625 ]$ Answer: C -Conrad and Dobler are the two top salesmen in an office that sells personal computers and security systems. The tables below show their sales figures for July and August. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { July Sales (\$) }$ Personal Computer Security System Conrad 25,000 44,000 Dobler 14,000 21,000 $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\text { August Sales (\$) }$ Personal Computers Security Systems Conrad 22,000 45,000 Dobler 18,000 27,000 Write the matrix which shows the increase (decrease) in sales from July to August.

Solve for x. - $\left| \begin{array} { l l } x & 3 \\2 & 1\end{array} \right| = 3$

Either the dimensions of two matrices or the matrices themselves are given. Find the dimensions of the product AB and the product BA. If either is not defined, say so. - $A = \left[ \begin{array} { l l } 3 & 4\end{array} \right] ; B = \left[ \begin{array} { r } - 3 \\6\end{array} \right]$