Q102 / 180 · COMEDK 2025 (10 May Shift 2) · Mathematics
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ₹ 500.
The cost of 1 kg onion, 2 kg wheat and 3 kg rice is ₹ 300 .
The cost of 6 kg onion, 2 kg wheat and 3 kg rice is ₹ 575 .
The above situation can be represented in matrix form as $\mathrm{AX}=\mathrm{B}$. Then $\left|5 A^{-1}\right|=$
- Correct answerA5
- B1
- C25
- D125
Solution
Let $x, y, z$ be the cost per kg of onion, wheat, and rice respectively. The given system of linear equations is:
$4x + 3y + 2z = 500$
$x + 2y + 3z = 300$
$6x + 2y + 3z = 575$
The matrix $A$ is given by:
$A = \begin{pmatrix} 4 & 3 & 2 \\ 1 & 2 & 3 \\ 6 & 2 & 3 \end{pmatrix}$
Calculate the determinant $|A|$:
$|A| = 4(2 \times 3 - 3 \times 2) - 3(1 \times 3 - 3 \times 6) + 2(1 \times 2 - 2 \times 6)$
$|A| = 4(6 - 6) - 3(3 - 18) + 2(2 - 12)$
$|A| = 4(0) - 3(-15) + 2(-10)$
$|A| = 0 + 45 - 20 = 25$
We need to find $|5A^{-1}|$. Using the property $|kA| = k^n |A|$ where $n$ is the order of the matrix:
$|5A^{-1}| = 5^3 |A^{-1}| = 125 \times \dfrac{1}{|A|}$
$|5A^{-1}| = 125 \times \dfrac{1}{25} = 5$
Answer: 5
$4x + 3y + 2z = 500$
$x + 2y + 3z = 300$
$6x + 2y + 3z = 575$
The matrix $A$ is given by:
$A = \begin{pmatrix} 4 & 3 & 2 \\ 1 & 2 & 3 \\ 6 & 2 & 3 \end{pmatrix}$
Calculate the determinant $|A|$:
$|A| = 4(2 \times 3 - 3 \times 2) - 3(1 \times 3 - 3 \times 6) + 2(1 \times 2 - 2 \times 6)$
$|A| = 4(6 - 6) - 3(3 - 18) + 2(2 - 12)$
$|A| = 4(0) - 3(-15) + 2(-10)$
$|A| = 0 + 45 - 20 = 25$
We need to find $|5A^{-1}|$. Using the property $|kA| = k^n |A|$ where $n$ is the order of the matrix:
$|5A^{-1}| = 5^3 |A^{-1}| = 125 \times \dfrac{1}{|A|}$
$|5A^{-1}| = 125 \times \dfrac{1}{25} = 5$
Answer: 5
Hence, the correct answer is option: 'A'.