Answer to Question #309830 in Linear Algebra for Zany

Question

 Doctors have become increasingly concerned about the sodium intake in the U.S. diet. Recommendations by the American Medical Association indicate that most individuals should not exceed 2400 mg of sodium per day. Liza ate 1 slice of pizza, 1 serving of ice cream, and 1 glass of soda for a total of 1030 mg of sodium. David ate 3 slices of pizza, no ice cream, and 2 glasses of soda for a total of 2420 mg of sodium. Melinda ate 2 slices of pizza, 1 serving of ice cream, and 2 glasses of soda for a total of 1910 mg of sodium. How much sodium is in one serving of each item?

Restriction: Use matrix inversion to answer to this question)

Answer 1

Let $x_1,x_2,x_3$ be the amount of sodium in 1 slice pizza, 1 ice-cream and 1 soda respectively. We have

$\left\{ \begin{array}{c} x_1+x_2+x_3=1030\\ 3x_1+2x_3=2420\\ 2x_1+x_2+2x_3=1910\\\end{array} \right. \Rightarrow \left( \begin{matrix} 1& 1& 1\\ 3& 0& 1\\ 2& 1& 2\\\end{matrix} \right) \left( \begin{array}{c} x_1\\ x_2\\ x_3\\\end{array} \right) =\left( \begin{array}{c} 1030\\ 2420\\ 1910\\\end{array} \right) \Rightarrow \\\Rightarrow \left( \begin{array}{c} x_1\\ x_2\\ x_3\\\end{array} \right) =\left( \begin{matrix} 1& 1& 1\\ 3& 0& 1\\ 2& 1& 2\\\end{matrix} \right) ^{-1}\left( \begin{array}{c} 1030\\ 2420\\ 1910\\\end{array} \right)$

Find $\left( \begin{matrix} 1& 1& 1\\ 3& 0& 1\\ 2& 1& 2\\\end{matrix} \right) ^{-1}$ with Gauss method:

$\left[ \begin{matrix} 1& 1& 1\\ 3& 0& 1\\ 2& 1& 2\\\end{matrix}\,\,\begin{matrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\\\end{matrix} \right] ~\left[ \begin{matrix} 1& 1& 1\\ 0& -3& -2\\ 0& -1& 0\\\end{matrix}\,\,\begin{matrix} 1& 0& 0\\ -3& 1& 0\\ -2& 0& 1\\\end{matrix} \right] ~\\~\left[ \begin{matrix} 1& 1& 1\\ 0& -3& -2\\ 0& 0& 2/3\\\end{matrix}\,\,\begin{matrix} 1& 0& 0\\ -3& 1& 0\\ -1& -1/3& 1\\\end{matrix} \right] ~\\~\left[ \begin{matrix} 1& 1& 1\\ 0& -3& -2\\ 0& 0& 1\\\end{matrix}\,\,\begin{matrix} 1& 0& 0\\ -3& 1& 0\\ -1.5& -0.5& 1.5\\\end{matrix} \right] ~\\~\left[ \begin{matrix} 1& 1& 0\\ 0& -3& 0\\ 0& 0& 1\\\end{matrix}\,\,\begin{matrix} 2.5& 0.5& -1.5\\ -6& 0& 3\\ -1.5& -0.5& 1.5\\\end{matrix} \right] ~\\~\left[ \begin{matrix} 1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\\\end{matrix}\,\,\begin{matrix} 2.5& 0.5& -1.5\\ 2& 0& -1\\ -1.5& -0.5& 1.5\\\end{matrix} \right] ~\\~\left[ \begin{matrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\\\end{matrix}\,\,\begin{matrix} 0.5& 0.5& -0.5\\ 2& 0& -1\\ -1.5& -0.5& 1.5\\\end{matrix} \right]$

from which the inverse matrix is

$\left[ \begin{matrix} 0.5& 0.5& -0.5\\ 2& 0& -1\\ -1.5& -0.5& 1.5\\\end{matrix} \right]$

Then

$\left( \begin{array}{c} x_1\\ x_2\\ x_3\\\end{array} \right) =\left( \begin{matrix} 1& 1& 1\\ 3& 0& 1\\ 2& 1& 2\\\end{matrix} \right) ^{-1}\left( \begin{array}{c} 1030\\ 2420\\ 1910\\\end{array} \right) =\\=\left( \begin{matrix} 0.5& 0.5& -0.5\\ 2& 0& -1\\ -1.5& -0.5& 1.5\\\end{matrix} \right) \left( \begin{array}{c} 1030\\ 2420\\ 1910\\\end{array} \right) =\left( \begin{array}{c} 770\\ 150\\ 110\\\end{array} \right)$

This means 1 slice pizza is 770 mg, 1 ice-cream is 150 mg, 1 soda is 110 mg.