In $$a multi$-$component engineering system, the strength requirements for various materials $($steel$,$concrete, wood$)$are interconnected. The strengths of $$the materials are given by $$the following

system of $$equations:

$3$x$-$y$+2$z$=1500$

$2$x$+2$y$-$z$=1200$

x$+3$y$+2$z$=1600$

Additionally, there is $$a cost factor associated with each material. The costs per ton are:

Steel: $$4000$

Concrete: $$3000$

Wood: $$1000$

The total cost of $$the materials is $$represented by $$the dot product of $$the material amounts vector x$,$y$,$z $$and the cost vector.

Use any matrix method to $$find the optimal combination of $$materials that meets the strength requirements while minimizing the cost.