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The problem gives a matrix showing the number of units of three products (P, Q, and R) that can be processed per hour on three machines (A, B, and C). The matrix is:

\[
\begin{bmatrix}
P & 10 & 12 & 15 \\
Q & 13 & 11 & 20 \\
R & 16 & 18 & 14 \\
\end{bmatrix}
\]

The goal is to determine how many units of each product can be produced, given the available hours for machines A, B, and C, which are 54, 46, and 48 hours respectively.

### Step 1: Matrix Setup

We can represent this as a matrix multiplication problem. Let matrix \( M \) represent the number of units processed per hour:

\[
M = \begin{bmatrix}
10 & 12 & 15 \\
13 & 11 & 20 \\
16 & 18 & 14 \\
\end{bmatrix}
\]

Let the available hours for each machine be represented as vector \( H \):

\[
H = \begin{bmatrix}
54 \\
46 \\
48 \\
\end{bmatrix}
\]

### Step 2: Matrix Multiplication

To find the total number of units of each product that can be produced, we multiply matrix \( M \) by vector \( H \):

\[
\text{Result} = M \times H
\]

The multiplication is:

\[
\text{Result} = \begin{bmatrix}
10 & 12 & 15 \\
13 & 11 & 20 \\
16 & 18 & 14 \\
\end{bmatrix}
\times
\begin{bmatrix}
54 \\
46 \\
48 \\
\end{bmatrix}
\]

I will now compute this matrix multiplication to determine the total units for each product.The total number of units of each product that can be produced is as follows:

- **Product P**: 1812 units
- **Product Q**: 2168 units
- **Product R**: 2364 units

These values are obtained by multiplying the matrix of production rates by the available hours for each machine.

Would you like more details on this or have any questions?

Here are 5 related questions to expand on this topic:
1. How does changing the available hours for the machines affect the total production?
2. What is the significance of matrix multiplication in this context?
3. Can we use this matrix to determine production rates if one machine breaks down?
4. How would we solve this problem if more machines were added?
5. What if we wanted to maximize production of only one product? How would we adjust the machine hours?

**Tip:** In matrix multiplication, always ensure that the number of columns in the first matrix matches the number of rows in the second matrix to perform the operation successfully.

The following matrix gives the number of units of three products (P, Q, and R) that can be processed per hour on three machines (A, B, and C):
Determine, by using matrix algebra, how many units of each product can be produced, if the hours available on machines A, B, and C are 54, 46, and 48 respectively.

Learn how to use matrix multiplication to determine the production output of three products (P, Q, and R) on machines A, B, and C. Given the hours available on the machines (54, 46, and 48), this problem shows a step-by-step matrix algebra solution suitable for Grades 11-12.

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