Vinayak runs a bakery shop. He sells three items: Sandwiches (₹ per unit), Fruit juices (₹ per…
Vinayak runs a bakery shop. He sells three items: Sandwiches (₹$x$ per unit), Fruit juices (₹$y$ per unit) and Cookies (₹$z$ per unit). The sales revenue over three days are 37, 26 and 37 respectively. The entire information is given below as matrix equation.
$\begin{pmatrix} 2 & 3 & 1 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 37 \\ 26 \\ 37 \end{pmatrix}$
Consider $A = \begin{pmatrix} 2 & 3 & 1 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{pmatrix}$ and $|A| = 17$. Find the price per unit for each item using matrix method.
| Sandwiches | Fruit Juices | Cookies | |||
|---|---|---|---|---|---|
| D1 | 2 | 3 | 1 | $x$ | 37 |
| D2 | 1 | 2 | 3 | $y$ | 26 |
| D3 | 3 | 1 | 1 | $z$ | 37 |
Answer
Answer
AIWritten by AI (gemini-2.5-pro) - it can contain mistakes.
The system in matrix form is $AX = B$ where
$A = \begin{pmatrix} 2 & 3 & 1 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, $B = \begin{pmatrix} 37 \\ 26 \\ 37 \end{pmatrix}$, and $|A| = 17$.
Cofactors of matrix $A$:
$C_{11} = 2 - 3 = -1$, $C_{12} = -(1 - 9) = 8$, $C_{13} = 1 - 6 = -5$
$C_{21} = -(3 - 1) = -2$, $C_{22} = 2 - 3 = -1$, $C_{23} = -(2 - 9) = 7$
$C_{31} = 9 - 2 = 7$, $C_{32} = -(6 - 1) = -5$, $C_{33} = 4 - 3 = 1$
$\text{adj}(A) = \begin{pmatrix} -1 & -2 & 7 \\ 8 & -1 & -5 \\ -5 & 7 & 1 \end{pmatrix}$
$X = A^{-1}B = \frac{1}{|A|} \text{adj}(A) B = \frac{1}{17} \begin{pmatrix} -1 & -2 & 7 \\ 8 & -1 & -5 \\ -5 & 7 & 1 \end{pmatrix} \begin{pmatrix} 37 \\ 26 \\ 37 \end{pmatrix}$
$x = \frac{1}{17}(-37 - 52 + 259) = \frac{170}{17} = 10$
$y = \frac{1}{17}(296 - 26 - 185) = \frac{85}{17} = 5$
$z = \frac{1}{17}(-185 + 182 + 37) = \frac{34}{17} = 2$
Therefore, the price per unit is:
Sandwiches ($x$) = ₹10
Fruit juices ($y$) = ₹5
Cookies ($z$) = ₹2
From ISC 2026 Mathematics Paper 1, question 7.