Question

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. (6 Marks)

Answer 1

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

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

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

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

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

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

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

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

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

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

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

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

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