Question

A company produces three products P, Q and R using raw materials A, B and C. One unit of P requires 1, 2 and 3 units of A, B and C respectively. One unit of Q requires 2, 3 and 2 units of A, B and C respectively. One unit of R requires 1, 2 and 2 units of A, B and C respectively. The number of units available for raw material A, B and C are 8, 14 and 13 units respectively. Using the matrix method, determine the number of units of each product to produce in order to utilize completely the available resources.

Answer 1

$\def\arraystretch{1.5} \begin{array}{c|c:c:c} & A(8) & B(14) & C(13)\\ \hline P & 1& 2& 3\\ \hdashline Q& 2 & 3& 2\\ \hdashline R & 1& 2 & 2 \end{array}$

$A=\begin{pmatrix} 1& 2 &1 \\ 2& 3 & 2\\ 3& 2 & 2 \end{pmatrix},$

$B=\begin{pmatrix} 8 \\ 14\\ 13 \end{pmatrix},$

$X=\begin{pmatrix} P \\ Q\\ R \end{pmatrix}.$

$|A|=\begin{vmatrix} 1& 2 &1 \\ 2& 3 & 2\\ 3& 2 & 2 \end{vmatrix}=1\cdot(6-4)-2\cdot(4-6)+1\cdot(4-9)=1\neq 0.$

$a_{11}=\begin{vmatrix} 3&2 \\ 2 & 2 \end{vmatrix}=2,$ $a_{12}=-\begin{vmatrix} 2&2 \\ 3 & 2 \end{vmatrix}=2,$ $a_{13}=\begin{vmatrix} 2&3 \\ 3 & 2 \end{vmatrix}=-5,$

$a_{21}=-\begin{vmatrix} 2&1 \\ 2 & 2 \end{vmatrix}=-2,$ $a_{22}=\begin{vmatrix} 1&1 \\ 3 & 2 \end{vmatrix}=-1,$ $a_{23}=-\begin{vmatrix} 1&2 \\ 3 & 2 \end{vmatrix}=4,$

$a_{31}=\begin{vmatrix} 2&1 \\ 3 & 2 \end{vmatrix}=1,$ $a_{32}=-\begin{vmatrix} 1&1 \\ 2 & 2 \end{vmatrix}=0,$ $a_{33}=\begin{vmatrix} 1&2 \\ 2 & 3 \end{vmatrix}=-1.$

$adjA={\begin{pmatrix} 2 & 2 &-5 \\ -2 & -1 &4 \\ 1& 0& -1 \end{pmatrix}}^{T}={\begin{pmatrix} 2& -2 &1 \\ 2& -1& 0\\ -5&4 & -1 \end{pmatrix}}.$

$A^{-1}=\frac{adjA}{|A|}=\begin{pmatrix} 2& -2 &1 \\ 2& -1& 0\\ -5&4 & -1 \end{pmatrix}.$

$X=A^{-1}B.$

$\begin{pmatrix} P \\ Q\\ R \end{pmatrix}=\begin{pmatrix} 2& -2 &1 \\ 2& -1& 0\\ -5&4 & -1 \end{pmatrix}\cdot \begin{pmatrix} 8 \\ 14\\ 13 \end{pmatrix}=\begin{pmatrix} 16-28+13 \\ 16-14\\ -40+56-13 \end{pmatrix}=\begin{pmatrix} 1 \\ 2\\ 3 \end{pmatrix}.$

$P=1, Q=2, R=3.$