Inverse Matrices: A Food Calculator

The forward problem was: eat amounts v of foods A, B, C and compute the nutrient totals Nv. This page solves the inverse problem: first choose the nutrient totals t you want for carbohydrate, fat, and protein, then ask how much of each food should be eaten. That is, solve Nv = t for the unknown v. When N is invertible, the answer is unique:

Nv = t ⇒ v = N⁻¹t, N⁻¹N = NN⁻¹ = I.

1. The nutrient matrix N — three food labels, side by side

Edit any entry (grams per 100 g of food). Column = food, row = nutrient.

g per 100 g	Food A	Food B	Food C
Carbohydrate
Fat
Protein

Enter a number from 0 to 100 in steps of 0.5 in every cell. Calculations resume when all entries are valid.

N =

N_1,1N_1,2N_1,3 N_2,1N_2,2N_2,3 N_3,1N_3,2N_3,3

entries N_i,j (row i = nutrient, column j = food)

2. The target vector t — the nutrients you want to eat

Slide to choose the desired total grams of each nutrient for the meal.

Carbohydrate

Fat

Protein

t =

t₁ t₂ t₃

entries t_i (row i = nutrient)

3. Does N⁻¹ exist? — the determinant decides

A 3×3 matrix is invertible exactly when det N ≠ 0, where

det N = N_1,1(N_2,2N_3,3 − N_2,3N_3,2)
det N = − N_1,2(N_2,1N_3,3 − N_2,3N_3,1)
det N = + N_1,3(N_2,1N_3,2 − N_2,2N_3,1).

det N = 0 — N is singular.
Therefore, N⁻¹ does not exist. Depending on t, the system Nv = t has either no solution or infinitely many solutions.

Since det N ≠ 0, the inverse exists:

N⁻¹ =

1 / det N

adj(N)

Here adj(N) is the adjugate matrix of N.

4. The solution v = N⁻¹t — how much of each food to eat

1 / det N

adj(N)

v = N⁻¹t

5. Check: Nv = t — undoing the inverse

Feed the computed amounts back into the forward problem. Multiplying by N undoes multiplying by N⁻¹: N(N⁻¹t) = (NN⁻¹)t = It = t. The target vector is recovered exactly.

1. The nutrient matrix N — three food labels, side by side

2. The target vector t — the nutrients you want to eat

3. Does N−1 exist? — the determinant decides

4. The solution v = N−1t — how much of each food to eat

5. Check: Nv = t — undoing the inverse

3. Does N⁻¹ exist? — the determinant decides

4. The solution v = N⁻¹t — how much of each food to eat