Question

Three students, Neha, Rani, and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads, and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads, and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads, and 3 erasers.
Based upon the above information, answer the following questions:

(ii) Find \( |A| \) and confirm if it is possible to find \( A^{-1} \).

Answer 1

The matrix \( A \) is: \[ A = \begin{pmatrix} 4 & 3 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 3 \end{pmatrix} \] We will compute the determinant of matrix \( A \), \( |A| \), using cofactor expansion: \[ |A| = 4 \begin{vmatrix} 4 & 6 \\ 2 & 3 \end{vmatrix} - 3 \begin{vmatrix} 2 & 6 \\ 6 & 3 \end{vmatrix} + 2 \begin{vmatrix} 2 & 4 \\ 6 & 2 \end{vmatrix} \] First, calculate the 2x2 minors: \[ \begin{vmatrix} 4 & 6 \\ 2 & 3 \end{vmatrix} = (4)(3) - (6)(2) = 12 - 12 = 0 \] \[ \begin{vmatrix} 2 & 6 \\ 6 & 3 \end{vmatrix} = (2)(3) - (6)(6) = 6 - 36 = -30 \] \[ \begin{vmatrix} 2 & 4 \\ 6 & 2 \end{vmatrix} = (2)(2) - (4)(6) = 4 - 24 = -20 \] Substituting back: \[ |A| = 4(0) - 3(-30) + 2(-20) = 0 + 90 - 40 = 50 \] Since \( |A| = 50 \neq 0 \), it is possible to find \( A^{-1} \).