Question

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is INR 500.
The cost of 1 kg onion, 2 kg wheat and 3 kg rice is INR 300.
The cost of 6 kg onion, 2 kg wheat and 3 kg rice is INR 575.
The above situation can be represented in matrix form as \( AX = B \). Then \( |5A^{-1}| = \)

• A. 5
• B. 125
• C. 1
• D. 25

Hint:

When multiplying a matrix by a scalar, the determinant is multiplied by the scalar raised to the power of the matrix's order. For an \( n \times n \) matrix \( A \), \( |kA| = k^n |A| \).

Answer 1

The Correct Option is A

Step 1: Represent the system of equations in matrix form.
Let the cost of 1 kg onion, wheat, and rice be \( x \), \( y \), and \( z \) respectively. Then, the given system of equations can be written as:
\[ \begin{bmatrix} 4 & 3 & 2 1 & 2 & 3 6 & 2 & 3 \end{bmatrix} \begin{bmatrix} x y z \end{bmatrix} = \begin{bmatrix} 500 300 575 \end{bmatrix} \]
This is the matrix form \( AX = B \), where:
\[ A = \begin{bmatrix} 4 & 3 & 2 1 & 2 & 3 6 & 2 & 3 \end{bmatrix}, \quad X = \begin{bmatrix} x y z \end{bmatrix}, \quad B = \begin{bmatrix} 500 300 575 \end{bmatrix} \]

Step 2: Calculate the determinant of \( A \).
The determinant of \( A \), denoted \( |A| \), is calculated as:
\[ |A| = \begin{vmatrix} 4 & 3 & 2 1 & 2 & 3 6 & 2 & 3 \end{vmatrix} \]
We use cofactor expansion along the first row:
\[ |A| = 4 \begin{vmatrix} 2 & 3 2 & 3 \end{vmatrix} - 3 \begin{vmatrix} 1 & 3 6 & 3 \end{vmatrix} + 2 \begin{vmatrix} 1 & 2 6 & 2 \end{vmatrix} \]
Calculating the 2x2 determinants:
\[ \begin{vmatrix} 2 & 3 2 & 3 \end{vmatrix} = 0, \quad \begin{vmatrix} 1 & 3 6 & 3 \end{vmatrix} = -15, \quad \begin{vmatrix} 1 & 2 6 & 2 \end{vmatrix} = -10 \]
Thus, \[ |A| = 4(0) - 3(-15) + 2(-10) = 0 + 45 - 20 = 25 \]

Step 3: Find \( |5A^{-1}| \).
We are asked to find \( |5A^{-1}| \). Using the property of determinants:
\[ |kA| = k^n |A| \quad \text{(where \( n \) is the order of the matrix)} \]
Since \( A \) is a 3x3 matrix, we have: \[ |5A| = 5^3 |A| = 125 |A| \]
Thus, \[ |A^{-1}| = \frac{1}{|A|} \]
and therefore: \[ |5A^{-1}| = 5^3 \times \frac{1}{|A|} = 125 \times \frac{1}{25} = 5 \]

Step 4: Conclusion.
Thus, the value of \( |5A^{-1}| \) is \( 5 \), and the correct answer is option (A).