## Chickens and Eggs

A check en farm produces chickens and eggs. Chickens can either lay eggs or hatch them. A chicken that lays eggs produces an average of 12 eggs a month and a chicken that hatches eggs hatches an average of 4 eggs a month. We can illustrate this with matrices. We can construct an input matrix and an output matrix.
Suppose we start with 1 chicken and 0 eggs. The output will be 1 chicken and 12 eggs.
Suppose we start with 1 chicken and 4 eggs. The output will be 5 chickens (the original plus 4 hatched from the eggs) and 0 eggs.
The input and output matrices are
$I=\left( \begin{array}{cc} 1 & 0 \\ 1 & 4 \end{array} \right), \: O=\left( \begin{array}{cc} 1 & 12 \\ 5 & 0 \end{array} \right)$
.
Let
$x_1$
be the number of laying chickens and let
$x_2$
be the number of hatching chickens. Hence
$(x_1, x_2)\left( \begin{array}{cc} 1 & 0 \\ 1 & 4 \end{array} \right)=(x_1+x_2, 4x_2)$
. There are now
$x_1+x_2$
chickens and
$4x_2$
Suppose we are told that we have 3 chickens and 8 eggs ready for hatching. Then
$(x_1,x_2)\left( \begin{array}{cc} 1 & 0 \\ 1 & 4 \end{array} \right)=(3,8)$
.
Hence
$(x_1,x_2)=(3,8){\left( \begin{array}{cc} 1 & 0 \\ 1 & 4 \end{array} \right)}^{-1}=(3,8){\left( \begin{array}{cc} 1 & 0 \\ -1/4 & 1/4 \end{array} \right)}=(1,2)$
.
Then 1 chicken is used for laying eggs and 2 chickens used for hatching.
$(1, 2)\left( \begin{array}{cc} 1 & 12 \\ 5 & 4 \end{array} \right)=(11, 12)$
.
The farmer now has 11 chickens and 12 eggs.