I dont want AI based answer, chegg expert please solve this through matrices.

Question $3$:

A company produces three products each of which mustg be prcessed through one department. Table $3\mathrm{.}6$ summarizes the labour hours and raw material requirement of each product. Each month there are $1500$ labour hours and $3800$ pounds of the raw material. If combined the monthly production for the three products should equal $500$ units, determine wether there are any combination of the three product which would exhaust the monthly availabilities of labour and raw material and meet the production goal of $500$ units.

$\backslash$table$[[$Department$,$Product $1,$Product $2,$Product $3],[$A$,3,2,4],[$B$,10,8,6]]$