## Chickens and Eggs

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 \[4x_2\]

eggs ready for hatching.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.