Question

Kiran purchased 3 pencils, 2 notebooks and one pen for INR 41. From the same shop Manasa purchased 2 pencils, one notebook and 2 pens for INR 29, while Shreya purchased 3 pencils, 2 notebooks and 2 pens for INR 44. The above situation can be represented in matrix form as \( AX = B \). Then \( \text{adj}(A) \) is equal to

• A. 9
• B. -9
• C. -1
• D. 1

Hint:

For an \( n \times n \) matrix, \( |\text{adj}(A)| = |A|^{n-1} \).

Answer 1

The Correct Option is D

Step 1: Form equations from given data.
Let pencil = \( x \), notebook = \( y \), pen = \( z \)
\[ 3x + 2y + z = 41 \]
\[ 2x + y + 2z = 29 \]
\[ 3x + 2y + 2z = 44 \]

Step 2: Write matrix form \( AX = B \).
\[ A = \begin{bmatrix} 3 & 2 & 1 2 & 1 & 2 3 & 2 & 2 \end{bmatrix} \]

Step 3: Find determinant of matrix \( A \).
\[ |A| = \begin{vmatrix} 3 & 2 & 1 2 & 1 & 2 3 & 2 & 2 \end{vmatrix} \]
Expand along first row:
\[ = 3 \begin{vmatrix} 1 & 2 2 & 2 \end{vmatrix} - 2 \begin{vmatrix} 2 & 2 3 & 2 \end{vmatrix} + 1 \begin{vmatrix} 2 & 1 3 & 2 \end{vmatrix} \]

Step 4: Calculate minors.
\[ = 3(1\cdot2 - 2\cdot2) - 2(2\cdot2 - 3\cdot2) + (2\cdot2 - 3\cdot1) \]
\[ = 3(2 - 4) - 2(4 - 6) + (4 - 3) \]
\[ = 3(-2) - 2(-2) + 1 \]
\[ = -6 + 4 + 1 = -1 \]

Step 5: Use property of adjoint.
\[ A \cdot \text{adj}(A) = |A| I \]

Step 6: Determinant of adjoint matrix.
\[ |\text{adj}(A)| = |A|^{n-1} \]
Here \( n = 3 \), so:
\[ |\text{adj}(A)| = (-1)^{2} = 1 \]

Step 7: Final Answer.
\[ \boxed{1} \]