Let's solve this problem step by step:
### Step 1: Represent the quantities ordered by each department in matrix form
Each row of the matrix corresponds to a department, and each column corresponds to a product.
[tex]\[
\text{Quantity Matrix} =
\begin{bmatrix}
8 & 5 & 4 & 3 & 6 \\
6 & 3 & 3 & 2 & 9 \\
4 & 7 & 1 & 0 & 10 \\
0 & 3 & 6 & 6 & 6
\end{bmatrix}
\][/tex]
### Step 2: Represent the prices from two suppliers
The prices from both suppliers for the ordered products:
[tex]\[
\text{Price from Supplier A} = [2, 1, 5, 3, 1]
\][/tex]
[tex]\[
\text{Price from Supplier B} = [3, 1, 3, 3, 1]
\][/tex]
### Step 3: Calculate the cost for each department from each supplier
We need to multiply the quantity matrix by the price vectors to get the cost for each department.
For Supplier A:
[tex]\[
\text{Cost}_{A} = \text{Quantity Matrix} \times \text{Price from Supplier A}
\][/tex]
[tex]\[
=
\begin{bmatrix}
8 & 5 & 4 & 3 & 6 \\
6 & 3 & 3 & 2 & 9 \\
4 & 7 & 1 & 0 & 10 \\
0 & 3 & 6 & 6 & 6
\end{bmatrix}
\times
\begin{bmatrix}
2 \\
1 \\
5 \\
3 \\
1
\end{bmatrix}
\][/tex]
The result from doing this operation is:
[tex]\[
\text{Cost}_A = [56, 45, 30, 57]
\][/tex]
For Supplier B:
[tex]\[
\text{Cost}_{B} = \text{Quantity Matrix} \times \text{Price from Supplier B}
\][/tex]
[tex]\[
=
\begin{bmatrix}
8 & 5 & 4 & 3 & 6 \\
6 & 3 & 3 & 2 & 9 \\
4 & 7 & 1 & 0 & 10 \\
0 & 3 & 6 & 6 & 6
\end{bmatrix}
\times
\begin{bmatrix}
3 \\
1 \\
3 \\
3 \\
1
\end{bmatrix}
\][/tex]
The result from doing this operation is:
[tex]\[
\text{Cost}_B = [56, 45, 32, 45]
\][/tex]
### Step 4: Calculate the total cost for all departments from each supplier.
Adding the costs for each department for Supplier A:
[tex]\[
\text{Total Cost}_A = 56 + 45 + 30 + 57 = 188
\][/tex]
Adding the costs for each department for Supplier B:
[tex]\[
\text{Total Cost}_B = 56 + 45 + 32 + 45 = 178
\][/tex]
### Step 5: Determine which supplier the company should purchase from
Since the total cost from Supplier B (178) is less than the total cost from Supplier A (188), the company should make the purchase from Supplier B.
### Final Answer:
- Supplier A Total Cost: [tex]\( \$188 \)[/tex]
- Supplier B Total Cost: [tex]\( \$178 \)[/tex]
The company should make the purchase from Supplier B.
