SOLUTION: A company manufactures three styles of kitchen cabinets and each style comes in two grades. Style 1 requires 4 square units of plywood and 14 hours to build. Style2 requires 5 squa

Question 30543: A company manufactures three styles of kitchen cabinets and each style comes in two grades. Style 1 requires 4 square units of plywood and 14 hours to build. Style2 requires 5 square units of plywood and 10 hours to build. Style 3 requires 3 square units of plywood and 8 hours to build. For grade A cabinets, the wood costs $26 per square unit and labour costs $35 per hour. For grade B, with cheaper material and less experienced cabinetmakers, the cost drops to $20 per square unit and $31 per hour. Display these figures in two matrices in such a way that their product shows the cost of materials and time required for each grade and style of cabinet. Calculate their product.
Answer by mbarugel(146) (Show Source):
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Hello!
Let's write the input requirements in a 3x2 matrix. The rows will represent each of the syles (3 rows); column 1 will represent the square units of plywood needed, and column will represent the hours of labor needed. We'll get this matrix:

Now, let's see how the "cost" matrix should be done. Recall that in matrix multiplication, you take each row in the first matrix and multiplicate it by each column in the 2nd matrix. A single *row* in the input matrix shows the amount of plywood and labor needed to build a given cabinet style. Therefore, in order to get the total cost when multiplicating, we should put the costs of plywood and labor arranged in *columns* in the cost matrix. SO the cost matrix will be:

Notice that each column shows the costs of plywood and labor for a given grade.
Let's check that the multiplication represents teh total costs for each grade and style. The matrix A*B will have 3 rows (number of rows in A) and 2 columns (number of columns in B). The element in position (1, 1) will be:
4*26 + 14*35 = 594
Notice that this shows the total cost for a cabinet of style 1, grade A. Now, the element in position (1, 2) will be
4*20 + 14*31 = 514
This element shows the total cost of a cabinet of style 1, grade B. The same reasoning can be made with the other elements of the multuiplication matrix. We finally get:

The element in position (i, j) shows the total cost of a cabinet of style i, grade j.
I hope this helps!
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