Q94 / 180 · COMEDK 2025 (10 May Shift 3) · Mathematics
Kiran purchased 3 pencils, 2 notebooks and one pen for ₹41.
From the same shop Manasa purchased 2 pencils, one notebook and 2 pens for ₹ 29, while Shreya purchased 3 pencils, 2 notebooks and 2 pens for ₹44.
The above situation can be represented in matrix form as $A X=B$.
Then $|\operatorname{adj} A|$ is equal to
- A$-1$
- Correct answerB$1$
- C$-9$
- D$9$
Solution
Coefficient matrix: $A = \begin{bmatrix} 3 & 2 & 1 \\ 2 & 1 & 2 \\ 3 & 2 & 2 \end{bmatrix}$
Applying $R_3 \to R_3 - R_1$:
$|A| = \begin{vmatrix} 3 & 2 & 1 \\ 2 & 1 & 2 \\ 0 & 0 & 1 \end{vmatrix}$
Expanding along $R_3$: $|A| = 1 \cdot (3 \times 1 - 2 \times 2) = 3 - 4 = -1$
For a $3 \times 3$ matrix: $|\text{adj } A| = |A|^{n-1} = |A|^2 = (-1)^2 = 1$
Applying $R_3 \to R_3 - R_1$:
$|A| = \begin{vmatrix} 3 & 2 & 1 \\ 2 & 1 & 2 \\ 0 & 0 & 1 \end{vmatrix}$
Expanding along $R_3$: $|A| = 1 \cdot (3 \times 1 - 2 \times 2) = 3 - 4 = -1$
For a $3 \times 3$ matrix: $|\text{adj } A| = |A|^{n-1} = |A|^2 = (-1)^2 = 1$
Hence, the correct answer is option: 'B'.