A furniture workshop produces three types of furniture - chairs, tables and beds each day. On a particular day the total number of furniture pieces produced is $\displaystyle 45$ . It was also found that production of beds exceeds that of chairs by $\displaystyle 8$ , while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using matrix method.

Let the numbers of chairs, tables and beds produced be $\displaystyle \mathrm{x}, \mathrm{y}$ and z respectively. \[\begin{aligned} & \therefore \mathrm{x}+\mathrm{y}+\mathrm{z}=45 ; \quad-\mathrm{x}+0 . \mathrm{y}+\mathrm{z}=8 ; \quad \mathrm{x}-2 \mathrm{y}+\mathrm{z}=0 \\ & \text { Let } A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ -1 & 0 & 1 \\ 1 & -2 & 1 \end{array}\right], X=\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \\ \mathrm{z} \end{array}\right], B=\left[\begin{array}{c} 45 \\ 8 \\ 0 \end{array}\right] \\ & |A|=1(0+2)-1(-1-1)+1(2-0)=6 \neq 0 \\ & \therefore A^{-1} \text { exists } \\ & \quad A X=B \Rightarrow X=A^{-1} B \\ & \operatorname{adj}(A)=\left[\begin{array}{ccc} 2 & -3 & 1 \\ 2 & 0 & -2 \\ 2 & 3 & 1 \end{array}\right] \\ & A^{-1}=\frac{1}{6}\left[\begin{array}{ccc} 2 & -3 & 1 \\ 2 & 0 & -2 \\ 2 & 3 & 1 \end{array}\right] \\ & \therefore\left[\begin{array}{l} \mathrm{x} \\ \mathrm{y} \\ \mathrm{z} \end{array}\right]=\frac{1}{6}\left[\begin{array}{ccc} 2 & -3 & 1 \\ 2 & 0 & -2 \\ 2 & 3 & 1 \end{array}\right]\left[\begin{array}{l} 45 \\ 8 \\ 0 \end{array}\right]=\left[\begin{array}{l} 11 \\ 15 \\ 19 \end{array}\right] \end{aligned} \] So, $\displaystyle \mathrm{x}=11, \mathrm{y}=15, \mathrm{z}=19$ Hence the numbers of chairs, tables and beds produced are $\displaystyle 11$, $\displaystyle 15$ and $\displaystyle 19$ respectively.