Math Problem Statement

A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce.

Type AType BType C

Folic acid (mg)

Choline (mg)

Inositol (mg)

(a) Find the inverse of the matrix below and use it to solve the remaining parts of this problem.

(b) How many ounces of each food should the nutritionist feed his laboratory rats if he wants their daily diet to contain 9 mg of folic acid, 18 mg of choline, and 8 mg of inositol?

Type A

Type B

Type C

Type A

Type B

Type C

Solution

import numpy as np

Define the nutrient matrix and find its inverse

nutrient_matrix = np.array([ [4, 1, 1], [2, 4, 2], [3, 1, 1] ])

Calculate the inverse of the nutrient matrix

inverse_matrix = np.linalg.inv(nutrient_matrix)

Part (b): Daily requirements for the first problem

requirements_b = np.array([9, 18, 8]) solution_b = np.dot(inverse_matrix, requirements_b)

Part (c): Daily requirements for the second problem

requirements_c = np.array([5, 4, 4]) solution_c = np.dot(inverse_matrix, requirements_c)

inverse_matrix, solution_b, solution_c(array([[ 1.00000000e+00, -1.38777878e-17, -1.00000000e+00], [ 2.00000000e+00, 5.00000000e-01, -3.00000000e+00], [-5.00000000e+00, -5.00000000e-01, 7.00000000e+00]]), array([1., 3., 2.]), array([1., 0., 1.]))(array([[ 1.00000000e+00, -1.38777878e-17, -1.00000000e+00], [ 2.00000000e+00, 5.00000000e-01, -3.00000000e+00], [-5.00000000e+00, -5.00000000e-01, 7.00000000e+00]]), array([1., 3., 2.]), array([1., 0., 1.]))success### Results

(a) Inverse of the nutrient matrix:

1 & -1.39 \times 10^{-17} & -1 \\ 2 & 0.5 & -3 \\ -5 & -0.5 & 7 \end{bmatrix}$$ (b) **Amounts of food required for 9 mg folic acid, 18 mg choline, and 8 mg inositol**: - Type A: **1 oz** - Type B: **3 oz** - Type C: **2 oz** (c) **Amounts of food required for 5 mg folic acid, 4 mg choline, and 4 mg inositol**: - Type A: **1 oz** - Type B: **0 oz** - Type C: **1 oz** Would you like a step-by-step explanation or help with related problems? Here are some related questions: 1. How can the inverse of a matrix be verified? 2. What if the nutrient requirements are different? Can we generalize the solution process? 3. How is matrix multiplication used in solving such problems? 4. Can the inverse matrix be interpreted in terms of nutrient contributions? 5. How would the solution change if one type of food was unavailable? **Tip:** Always check if a matrix is invertible by ensuring its determinant is non-zero before attempting to find its inverse!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Systems of Linear Equations

Formulas

Matrix inversion formula: A⁻¹ = adj(A)/det(A)
Matrix multiplication: Ax = b

Theorems

Invertible Matrix Theorem: A square matrix is invertible if and only if its determinant is non-zero.

Suitable Grade Level

Undergraduate (Mathematics, Linear Algebra)

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