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:

(iii) (b) Find \( A^2 - I \), where \( I \) is the identity matrix.

Hint:

Remember to check the determinant of \( A \) before attempting to find \( A^{-1} \), as \( A^{-1} \) exists only if \( |A| \neq 0 \).

Answer 1

The identity matrix \( I \) is: \[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \] We need to compute \( A^2 \). To do so, multiply \( A \) by itself: \[ A^2 = \begin{pmatrix} 4 & 3 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 3 \end{pmatrix} \begin{pmatrix} 4 & 3 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 3 \end{pmatrix} \] \[ A^2 = \begin{pmatrix} (4)(4)+(3)(2)+(2)(6) & (4)(3)+(3)(4)+(2)(2) & (4)(2)+(3)(6)+(2)(3) \\ (2)(4)+(4)(2)+(6)(6) & (2)(3)+(4)(4)+(6)(2) & (2)(2)+(4)(6)+(6)(3) \\ (6)(4)+(2)(2)+(3)(6) & (6)(3)+(2)(4)+(3)(2) & (6)(2)+(2)(6)+(3)(3) \end{pmatrix} \] Simplifying: \[ A^2 = \begin{pmatrix} 40 & 30 & 30 \\ 56 & 42 & 42 \\ 48 & 38 & 39 \end{pmatrix} \] Now subtract the identity matrix \( I \): \[ A^2 - I = \begin{pmatrix} 40 & 30 & 30 \\ 56 & 42 & 42 \\ 48 & 38 & 39 \end{pmatrix} - \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \] \[ A^2 - I = \begin{pmatrix} 39 & 30 & 30 \\ 56 & 41 & 42 \\ 48 & 38 & 38 \end{pmatrix} \]