Math Problem Statement

At a music festival, souvenir hats, t-shirts, and sweatshirts are sold. The cost of three hats, two t-shirts, and one sweatshirt is $140. Two hats, two t-shirts, and two sweatshirts cost $170. And the cost of one hat, three t-shirts, and two sweatshirts is $180. Find the prices of each individual item. Solve using inverse matrix.

Solution

To solve this problem using the inverse matrix method, we start by setting up a system of linear equations based on the given information.

Let's denote:

  • xx as the price of one hat
  • yy as the price of one t-shirt
  • zz as the price of one sweatshirt

The given information translates into the following system of equations:

3x + 2y + z = 140 \\ 2x + 2y + 2z = 170 \\ x + 3y + 2z = 180 \end{cases}$$ We can write this system in matrix form $$AX = B$$, where $$A$$ is the coefficient matrix, $$X$$ is the column matrix of the variables, and $$B$$ is the column matrix of the constants. Specifically: $$A = \begin{pmatrix} 3 & 2 & 1 \\ 2 & 2 & 2 \\ 1 & 3 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 140 \\ 170 \\ 180 \end{pmatrix}$$ To solve for $$X$$, we need to find the inverse of matrix $$A$$, denoted as $$A^{-1}$$, and then multiply it by matrix $$B$$: $$X = A^{-1} B$$ First, we compute the inverse of $$A$$. The inverse of a 3x3 matrix $$A$$ is given by: $$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$$ where $$\det(A)$$ is the determinant of $$A$$ and $$\text{adj}(A)$$ is the adjugate of $$A$$. ### Step 1: Compute the Determinant of $$A$$ $$\det(A) = 3 \begin{vmatrix} 2 & 2 \\ 3 & 2 \end{vmatrix} - 2 \begin{vmatrix} 2 & 2 \\ 1 & 2 \end{vmatrix} + 1 \begin{vmatrix} 2 & 2 \\ 1 & 3 \end{vmatrix}$$ $$\det(A) = 3 (2 \cdot 2 - 2 \cdot 3) - 2 (2 \cdot 2 - 2 \cdot 1) + 1 (2 \cdot 3 - 2 \cdot 1)$$ $$\det(A) = 3 (4 - 6) - 2 (4 - 2) + 1 (6 - 2)$$ $$\det(A) = 3 (-2) - 2 (2) + 1 (4)$$ $$\det(A) = -6 - 4 + 4 = -6$$ ### Step 2: Compute the Adjugate of $$A$$ The adjugate of $$A$$, $$\text{adj}(A)$$, is the transpose of the cofactor matrix of $$A$$. $$\text{adj}(A) = \begin{pmatrix} 2 & -2 & 2 \\ 4 & -5 & 5 \\ 2 & -4 & 4 \end{pmatrix}$$ ### Step 3: Compute $$A^{-1}$$ $$A^{-1} = \frac{1}{\det(A)} \text{adj}(A) = \frac{1}{-6} \begin{pmatrix} 2 & -2 & 2 \\ 4 & -5 & 5 \\ 2 & -4 & 4 \end{pmatrix}$$ $$A^{-1} = \begin{pmatrix} -\frac{1}{3} & \frac{1}{3} & -\frac{1}{3} \\ -\frac{2}{3} & \frac{5}{6} & -\frac{5}{6} \\ -\frac{1}{3} & \frac{2}{3} & -\frac{2}{3} \end{pmatrix}$$ ### Step 4: Multiply $$A^{-1}$$ by $$B$$ $$X = A^{-1} B = \begin{pmatrix} -\frac{1}{3} & \frac{1}{3} & -\frac{1}{3} \\ -\frac{2}{3} & \frac{5}{6} & -\frac{5}{6} \\ -\frac{1}{3} & \frac{2}{3} & -\frac{2}{3} \end{pmatrix} \begin{pmatrix} 140 \\ 170 \\ 180 \end{pmatrix}$$ $$X = \begin{pmatrix} -\frac{1}{3} \cdot 140 + \frac{1}{3} \cdot 170 - \frac{1}{3} \cdot 180 \\ -\frac{2}{3} \cdot 140 + \frac{5}{6} \cdot 170 - \frac{5}{6} \cdot 180 \\ -\frac{1}{3} \cdot 140 + \frac{2}{3} \cdot 170 - \frac{2}{3} \cdot 180 \end{pmatrix}$$ $$X = \begin{pmatrix} -\frac{140}{3} + \frac{170}{3} - \frac{180}{3} \\ -\frac{280}{3} + \frac{850}{6} - \frac{900}{6} \\ -\frac{140}{3} + \frac{340}{3} - \frac{360}{3} \end{pmatrix}$$ $$X = \begin{pmatrix} -46.67 + 56.67 - 60 \\ -93.33 + 141.67 - 150 \\ -46.67 + 113.33 - 120 \end{pmatrix}$$ $$X = \begin{pmatrix} -50 \\ -101.67 \\ -53.33 \end{pmatrix}$$ Thus, the prices of the hat, t-shirt, and sweatshirt are $$\boxed{-50}$$, $$\boxed{-101.67}$$, and $$\boxed{-53.33}$$ respectively. The negative values indicate a calculation error; let's recompute by ensuring the arithmetic is correctly applied.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Matrix Operations
Inverse Matrices

Formulas

Matrix multiplication
Determinant of a 3x3 matrix
Inverse of a 3x3 matrix

Theorems

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Suitable Grade Level

Advanced High School