Math Tip of the Week – A Matrix Problem

Hi Math folks!  So sorry I haven’t been blogging recently; I’ve been working on getting my tutoring schedule worked out and have just been trying to catch up after a whirlwind trip to visit both of my sons “up north”  🙂

Here’s an interesting problem that someone just sent me:

Determine the solution by setting up and solving the matrix equation.
A nut distributor wants to determine the nutritional content of various mixtures of pecans, cashews, and almonds. Her supplier has provided the following nutrition information:
Almonds. Cashews. Pecans
Protein. 26.2g/cup. 21.0g/cup. 10.1g/cup
Carbs. 40.2g/cup. 44.8g/cup. 14.3g/cup
Fat. 71.9g/cup. 63.5g/cup. 82.8g/cup

Her first mixture, protein blend, contains 6 cups of almonds, 3 cups of cashews, and 1 cup of pecans. Her second mixture, low fat mix, contains 3 cups almonds, 6 cups cashews, and 1 cup of pecans. Her third mixture, low carb mix contains 3 cups almonds, 1 cup cashews, and 6 cups pecans. Determine the amount of protein, carbs, and fats in a 1 cup serving of each of the mixtures.

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I have solved this by multiplying both of the matrices then dividing each element by 10 but that’s not the way I am supposed to solve this as there are no equations being set up.

Solution:

Sometimes we can just put the information we have into matrices and see how we are going to go from there. I knew to put the first group of data into a matrix with Almonds, Cashews, and Pecans as columns, and then put the second group of data into a matrix with information about Almonds, Cashews, and Pecans as rows. This way the columns of the first matrix lined up with the rows of the second matrix, and I could perform matrix multiplication. This way we get rid of the number of cups of Almonds, Cashews, and Pecans, which we don’t need:

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Then we can multiply the matrices (we can use a graphing calculator) since we want to end up with the amount of Protein, Carbs, and Fat in each of the mixtures. The product of the matrices consists of rows of Protein, Carbs, and Fat, and columns of the Protein, Low Fat, and Low Carb mixtures:

$$\displaystyle \require {cancel} \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\cancel{{\text{Almonds, Cashews and Pecans}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Protein, Low-Fat and Carb }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Protein, Low-Fat and Carb}\\\,\,\begin{array}{*{20}{c}} {\text{Protein}} \\ {\text{Carbs}} \\ {\text{Fat}} \end{array}\,\,\,\,\,\left[ {\begin{array}{*{20}{c}} {26.2} & {21} & {10.1} \\ {40.2} & {44.8} & {14.3} \\ {71.9} & {63.5} & {82.8} \end{array}} \right]\,\,\,\,\,\,\times \,\,\,\cancel{{\begin{array}{*{20}{c}} {\text{Almonds}} \\ {\text{Cashews}} \\ {\text{Pecans}} \end{array}}}\,\,\,\,\,\left[ {\begin{array}{*{20}{c}} 6 & 3 & 3 \\ 3 & 6 & 1 \\ 1 & 1 & 6 \end{array}} \right]\,\,\,\,\,\,\,=\,\,\,\begin{array}{*{20}{c}} {\text{Protein}} \\ {\text{Carbs}} \\ {\text{Fat}} \end{array}\,\,\,\,\left[ {\begin{array}{*{20}{c}} {230.3} & {214.7} & {160.2} \\ {389.9} & {403.7} & {251.2} \\ {704.7} & {679.5} & {776} \end{array}} \right]\end{array}$$

But we have to be careful, since these amounts are for 10 cups (add down to see we’ll get 10 cups for each mixture in the second matrix above). Also, notice how the cups unit “canceled out” when we did the matrix multiplication (grams/cup time cups = grams).

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To get the answers, we have to divide each answer by 10 to get grams per cup. The numbers in bold are our answers:

 Protein Blend Low Fat Mixture Low Carb Mixture Protein (grams) 230.3/10 = 23.03 214.7/10 = 21.47 160.2/10 = 16.02 Carbs (grams) 389.9/10 = 38.9 403.7/10=40.37 251.2/10 = 25.12 Fat (grams) 704.7/10 = 70.47 679.5/10 = 67.95 776/10 = 77.6

Learn more about matrices, how to use the graphing calculator with them,  and how to solve Systems of Equations using matrices in the The Matrix and Solving Systems with Matrices section!

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