MATRIX HELP

[mickdano]

Hello.. I'm not really good at word problems. This one has stumped me. Tutor for my school doesn't understand the process that my text gives either. So now I'm completely confused. All I need is "STEP BY STEP" on how to do it.. and I can teach myself. Anyone, please.

A supplier for the automobile industry manufactures car and truck frames at two different plants. How many hours should each plant be scheduled to operate to exactly fill each of the orders in the right table? The production rates (in frames per hour) for each plant are given below.

Plant A - 2/Car Frames, 7/Truck Frames
Plant B - 4/Car Frames, 4/Truck Frames
Orders - #1 824/ Car Frames, 1004/Truck Frames
#2 506/Car Frames, 1051/Truck Frames
#3 1096/Car Frames, 2036/Truck Frames

Complete the table

Orders 1 2 3
Plant A hours
Plant B hours

It breaks down and I have made it as far as

2 4 1 0
7 4 0 1

The rest is lost to me.

Thank you!

 

[mmm4444bot]

Complete the table

Orders 1 2 3
Plant A hours
Plant B hours

it breaks down and I have made it as far as

2 4 1 0
7 4 0 1

Your setup looks to me like finding the inverse of the coefficient matrix. Perhaps, your textbook writes matrix equations and solves them with matrix multiplication on both sides by the inverse.

I'm not sure what you mean, when you say "the rest is lost".

:?: Are you asking for help with the row operations, to find the inverse matrix ?

Here they are, step by step.

(1) Multiply row 1 by 0.5

(2) Add -7 times row 1 to row 2

(3) Multiply row 2 by -0.1

(4) Add -2 times row 2 to row 1

Columns 3 and 4 of the result comprise the inverse matrix.

If you write the order requirements as a 2x1 matrix, and then multiply that by the inverse above (using matrix multiplication), the resulting 2x1 matrix will be the solutions for A and B. You need to do this for each order, to complete the table.

Solving the matrix equation for the first one looks like this:

| -0.20  0.20 |  *  |  824 |  =  |  36.00 |
|  0.35 -0.10 |     | 1004 |     | 188.00 |

To fulfill order #1, Plant A operates 36 hours and Plant B operates 188 hours.

There are a few other methods, to solve for A and B. For example, since you're doing row operations anyway, they could be done on an augmented coefficient matrix leading directly to the solution, without the need for matrix multiplication.

Again, I don't know what your class is doing, aside from (apparently) using matrices to solve systems of equations. If you need more help, please try to be specific.

MY EDITS: Changed 1x1 to 2x1 throughout (Thanks wjm11)

 

[soroban]

Hello, mickdano!

It required a few readings, but I think I've assimilated the problem.
. . (Resistance is futile!)

A supplier for the automobile industry manufactures car and truck frames at two different plants.
How many hours should each plant be scheduled to operate to exactly fill each of the orders in the right table?
The production rates (in frames per hour) for each plant are given below.

. . \(\displaystyle \begin{array}{c||c|c|} & \text{cars} & \text{trucks} \\ \hline \hline \text{Plant A (per hour)} & 2 & 7 \\ \hline \text{Plant B (per hour)} & 4 & 4 \\ \hline \end{array} \qquad\qquad \begin{array}{c||c|c|} & \text{cars} & \text{trucks} \\ \hline \hline \text{Order 1} & 824 & 1004 \\ \hline \text{Order 2} & 506 & 1051 \\ \hline \text{Order 3} & 1096 & 2036 \\ \hline \end{array}\)


\(\displaystyle \text{Complete the table: }\quad\begin{array}{c||c|c|c|} \text{Orders} & 1 & 2 & 3 \\ \hline\hline \text{Plant A (hrs)} & & &\\ \hline \text{Plant B (hrs)} & & & \\ \hline \end{array}\)

This seemed to be a linear programming problem,
. . but it does not ask for a maximum or minimum quantity.

I finally saw that this is simply "a system of two linear equations" . . . three times.


\(\displaystyle \text{The first table says: }\;\begin{array}{c}\text{Plant A produces 2 cars and 7 trucks per hour.} \\ \text{Plant B produces 4 cars and 4 trucks per hour.} \end{array}\)


\(\displaystyle \text{Let }x\text{ = number of hours that Plant A operates.}\)
\(\displaystyle \text{Let }y\text{ = number of hours that Plant B operates.}\)


\(\displaystyle \text{Then the production of cars and trucks looks like this:}\)

. . \(\displaystyle \begin{array}{c||c|c|} & \text{cars} & \text{trucks} \\ \hline \hline \text{Plant A} & 2x & 7x \\ \hline \text{Plant B} & 4y & 4y \\ \hline \end{array}\)



\(\displaystyle \text{Let's fill Order 1.}\)
\(\displaystyle \text{The table looks like this: }\;\;\begin{array}{c||c|c|} & \text{cars} & \text{trucks} \\ \hline \hline \text{Plant A} & 2x & 7x \\ \hline \text{Plant B} & 4y & 4y \\ \hline \hline \text{Order 1} & 824 & 1004 \\ \hine\hline \end{array}\)

\(\displaystyle \text{Reading down the columns, we have these equations: }\;\begin{array}{ccc}2x + 4y &=& 824 \\ 7x + 4y &=& 1004 \end{array}\)

\(\displaystyle \text{Solve the system: }\;x = 36,\;y = 188\)


\(\displaystyle \text{And we have (so far): }\;\begin{array}{c||c|c|c|} \text{Orders} & 1 & 2 & 3 \\ \hline\hline \text{Plant A (hrs)} & 36 & &\\ \hline \text{Plant B (hrs)} & 188 & & \\ \hline \end{array}\)


Edit: corrected my blunder . . .

 

[mickdano]

Thank you so much!!

 

[mmm4444bot]

My prior statement that each system could be found using the augmented coefficient matrix is, on second thought, not the best way to go, if you're being taught to solve these systems using matrices, since it would require three sets of row operations.

Using the same inverse matrix on all three systems, however, only requires one set of row operations, plus the three multiplications (matrix multiplication is easy for a 2x2 times a 2x1).