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.

Question

A matrix with 2 rows and 2 columns, where row 1 is 2 and 3 and row 2 is 1 and 1, is multiplied by matrix with 2 rows and 1 column, where row 1 is x and row 2 is y, equals a matrix with 2 rows and 1 column, where row 1 is 40 and row 2 is 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.

Answer 1

Let's start by setting up the equations using matrices.

Let's assume the cost of a t-shirt is x and the cost of a notebook is y.

For Club A:
2t-shirts + 3notebooks = $40

For Club B:
1 t-shirt + 1 notebook = $16

We can write this as a matrix equation:

A * X = B

Where A is a 2x2 matrix:
| 2 3 |
| 1 1 |

X is a 2x1 matrix:
| x |
| y |

And B is a 2x1 matrix:
| 40 |
| 16 |

We need to find the values of x and y.

To solve the equation, we need to find the inverse of matrix A and multiply it on both sides of the equation.

A^-1 * A * X = A^-1 * B

Since A^-1 * A gives us the identity matrix, the equation simplifies to:

X = A^-1 * B

Let's calculate the inverse of matrix A:

A^-1 = (1 / (ad - bc)) * | d -b |
| -c a |

a = 2, b = 3, c = 1, d = 1

So, a^-1 = (1 / (2*1 - 3*1)) * | 1 -3 |
| -1 2 |

a^-1 = (1 / -1) * | 1 -3 |
| -1 2 |

a^-1 = | -1 3 |
| 1 -2 |

Now, let's calculate A^-1 * B:

| -1 3 | | 40 | | (-1*40) + (3*16) | | -40 + 48 | | 8 |
| 1 -2 | * | 16 | = | (1*40) + (-2*16) | = | 40 - 32 | = | 8 |

So, X = | 8 |
| 8 |

Therefore, the cost of a t-shirt is $8 and the cost of a notebook is $8.