A dietician prepares a diet specifying the amounts a patient should eat of the four basic food groups: group I, meats; group II, fruits and vegetables; group III, breads and starches; group IV, milkproducts. Amounts are given in "exchanges" that represent 1 oz (meat), 1/2 cup (fruits and vegetables), 1 slice (bread), 8 oz (milk), or other suitable measurements. Matrix X represents the numberof "exchanges" for breakfast in row 1, lunch in row 2, and dinner in row 3, for each food group, respectively. Matrix Y represents the amounts of fat in column 1, carbohydrates in column 2, andprotein in column 3. Matrix Z represents calories per exchange of fat in row 1, carbohydrates in row 2, and protein in row 3. Complete parts (a) through (c) below.X=Y=Z=1 1 1 12 2 2 331 3 36191 13 10 17 4613 141045C(a) Find the product matrix XY.XY =What do the entries of this matrix represent?Theof thegive the amounts of fat, carbohydrates, and protein, respectively, in each(b) Find the product matrix YZ.YZ=What do the entries represent?Thegroups.(c) Find the products (XY)Z and X(YZ) and verify that they are equal.(XY)Z =X(YZ) =What do the entries represent?Thegive the number of calories ingive the number of calories inof each of the food

Given three matrices. X=111122233133Y=6191131017413146Z=1045 (a) Find the product matrix XY…

Answered: X= Y= Z= 1 1 1 1 22 23 3 1 33 6 19 1 13…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A dietician prepares a diet specifying the amounts a patient should eat of the four basic food groups: group I, meats; group II, fruits and vegetables; group III, breads and starches; group IV, milk
products. Amounts are given in "exchanges" that represent 1 oz (meat), 1/2 cup (fruits and vegetables), 1 slice (bread), 8 oz (milk), or other suitable measurements. Matrix X represents the number
of "exchanges" for breakfast in row 1, lunch in row 2, and dinner in row 3, for each food group, respectively. Matrix Y represents the amounts of fat in column 1, carbohydrates in column 2, and
protein in column 3. Matrix Z represents calories per exchange of fat in row 1, carbohydrates in row 2, and protein in row 3. Complete parts (a) through (c) below.
X=
Y=
Z=
1 1 1 1
2 2 2 3
3
1 3 3
6
19
1 13 1
0 17 4
6
13 14
10
4
5
C
(a) Find the product matrix XY.
XY =
What do the entries of this matrix represent?
The
of the
give the amounts of fat, carbohydrates, and protein, respectively, in each
(b) Find the product matrix YZ.
YZ=
What do the entries represent?
The
groups.
(c) Find the products (XY)Z and X(YZ) and verify that they are equal.
(XY)Z =
X(YZ) =
What do the entries represent?
The
give the number of calories in
give the number of calories in
of each of the food

Expert Solution

Step 1

Given three matrices.

$X = [\begin{matrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 3 \\ 3 & 1 & 3 & 3 \end{matrix}] Y = [\begin{matrix} 6 & 1 & 9 \\ 1 & 13 & 1 \\ 0 & 17 & 4 \\ 13 & 14 & 6 \end{matrix}] Z = [\begin{matrix} 10 \\ 4 \\ 5 \end{matrix}]$

(a) Find the product matrix $X Y$

$X Y = [\begin{matrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 3 \\ 3 & 1 & 3 & 3 \end{matrix}] [\begin{matrix} 6 & 1 & 9 \\ 1 & 13 & 1 \\ 0 & 17 & 4 \\ 13 & 14 & 6 \end{matrix}] = [\begin{matrix} 6 + 1 + 0 + 13 & 1 + 13 + 17 + 14 & 9 + 1 + 4 + 6 \\ 12 + 2 + 0 + 39 & 2 + 26 + 34 + 42 & 18 + 2 + 8 + 18 \\ 18 + 1 + 0 + 39 & 3 + 13 + 51 + 42 & 27 + 1 + 12 + 18 \end{matrix}] = [\begin{matrix} 20 & 45 & 20 \\ 53 & 104 & 46 \\ 58 & 109 & 58 \end{matrix}]$