Fruit salad/Minerals and vitamins/Table and matrix/Linear system/Example

A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium and magnesium various fruits have (in milligram with respect to 100 gram of the fruit).

	apple	orange	grapes	banana
vitamin C	${}12$	${}53$	${}4$	${}9$
calcium	${}7$	${}40$	${}12$	${}5$
magnesium	${}6$	${}10$	${}8$	${}27$

My fruit salad today consists of the mentioned fruits with portions ${}{\begin{pmatrix}3\\2\\7\\6\end{pmatrix}}$ (meaning ${}300$ gram apple and so on). From that, one can calculate the total vitamin-C-amount, the calcium-amount and the magnesium-amount of the fruit salad, by simply multiplying for each fruit its portion with its specific amount, and summing up everything. The vitamin-C-amount of the complete fruit salad is thus

${}12\cdot 3+53\cdot 2+4\cdot 7+9\cdot 6=224\,.$

This operation is an example for how a matrix operates. The table yields immediately a ${}3\times 4$-matrix, namely ${}{\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}$, and the above calculation is realized by the matrix multiplication

${}{\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}{\begin{pmatrix}3\\2\\7\\6\end{pmatrix}}={\begin{pmatrix}224\\215\\256\end{pmatrix}}\,.$

One can also ask for a fruit salad which has certain amounts of vitamin C, calcium and magnesium, say ${}{\begin{pmatrix}180\\110\\140\end{pmatrix}}$. This leads to the linear system of linear equations in matrix form,

${}{\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}={\begin{pmatrix}180\\110\\140\end{pmatrix}}\,.$