Two student clubs were selling t-shirts and school notebooks to raise money for an upcoming school event. In the first few minutes, club A sold 2 t-shirts and 3 notebooks, and made $40. Club B sold 1 t-shirt and 1 notebook, for a total of $16.



Use matrices to solve the equation and determine the cost of a t-shirt and the cost of a notebook. Show or explain all necessary steps.

To determine the cost of a t-shirt and a notebook, we can set up a system of linear equations based on the information given. Then, we can solve this system using matrices.

### Step 1: Set Up the Equations
Let's denote:
- $ t $ as the cost of a t-shirt (in dollars)
- $ n $ as the cost of a notebook (in dollars)

From the problem, we have two equations:
1. **Club A**: $ 2t + 3n = 40 $
2. **Club B**: $ t + n = 16 $

### Step 2: Express the System as a Matrix Equation
We can express this system of equations in matrix form as follows:

\[
\begin{pmatrix}
2 & 3 \\
1 & 1
\end{pmatrix}
\begin{pmatrix}
t \\
n
\end{pmatrix}
=
\begin{pmatrix}
40 \\
16
\end{pmatrix}
\]

### Step 3: Solve the Matrix Equation
The matrix equation is $ A \mathbf{x} = \mathbf{b} $, where:

\[
A = \begin{pmatrix}
2 & 3 \\
1 & 1
\end{pmatrix}
, \quad \mathbf{x} = \begin{pmatrix}
t \\
n
\end{pmatrix}
, \quad \mathbf{b} = \begin{pmatrix}
40 \\
16
\end{pmatrix}
\]

To solve for $ \mathbf{x} $, we need to find the inverse of matrix $ A $ (denoted as $ A^{-1} $) and then multiply it by vector $ \mathbf{b} $:

\[
\mathbf{x} = A^{-1} \mathbf{b}
\]

### Step 4: Find the Inverse of Matrix $ A $
For a $ 2 \times 2 $ matrix $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $, the inverse is given by:

\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

Substituting the values from matrix $ A $:

\[
A^{-1} = \frac{1}{(2)(1) - (3)(1)} \begin{pmatrix} 1 & -3 \\ -1 & 2 \end{pmatrix}
\]

\[
A^{-1} = \frac{1}{2 - 3} \begin{pmatrix} 1 & -3 \\ -1 & 2 \end{pmatrix} = \frac{1}{-1} \begin{pmatrix} 1 & -3 \\ -1 & 2 \end{pmatrix}
\]

\[
A^{-1} = \begin{pmatrix} -1 & 3 \\ 1 & -2 \end{pmatrix}
\]

### Step 5: Multiply $ A^{-1} $ by $ \mathbf{b} $ to Find $ \mathbf{x} $
Now, multiply $ A^{-1} $ by $ \mathbf{b} $:

\[
\mathbf{x} = \begin{pmatrix} -1 & 3 \\ 1 & -2 \end{pmatrix} \begin{pmatrix} 40 \\ 16 \end{pmatrix}
\]

Perform the matrix multiplication:

\[
\mathbf{x} = \begin{pmatrix} -1 \cdot 40 + 3 \cdot 16 \\ 1 \cdot 40 - 2 \cdot 16 \end{pmatrix} = \begin{pmatrix} -40 + 48 \\ 40 - 32 \end{pmatrix} = \begin{pmatrix} 8 \\ 8 \end{pmatrix}
\]

### Step 6: Interpret the Solution
The solution vector $ \mathbf{x} = \begin{pmatrix} 8 \\ 8 \end{pmatrix} $ gives us:
- $ t = 8 $ (The cost of a t-shirt is $8)
- $ n = 8 $ (The cost of a notebook is $8)

### Conclusion
The cost of a t-shirt is $8, and the cost of a notebook is also $8.

Would you like any further details or have any questions?

### Relative Questions:
1. What are the steps to find the inverse of a $ 2 \times 2 $ matrix?
2. How do you interpret the solution of a system of linear equations using matrices?
3. What happens if the determinant of matrix $ A $ is zero?
4. Can you solve the same problem using another method, such as substitution or elimination?
5. How can matrix multiplication be verified in this context?
6. What are the properties of an inverse matrix?
7. How can the determinant be used to check if a matrix has an inverse?
8. Can this method be extended to solve larger systems of equations?

### Tip:
When solving linear equations with matrices, always check the determinant first to ensure the matrix is invertible.

Learn how to solve linear equations using matrices to find the cost of t-shirts and notebooks for two student clubs. This problem covers key concepts in linear algebra, including matrix multiplication and finding the inverse of a 2x2 matrix. Suitable for high school students.

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