Problem 39 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}\) and \(A_{3} B_{3} C_{3}\) are three three-digit numbers, each of which is divisible by \(k\), then \(\Delta=\left|\begin{array}{ccc}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|\) is (A) divisible by \(k\) (B) divisible by \(k^{2}\) (C) divisible by \(2 k\) (D) None of these

Step-by-Step Solution

Answer

The determinant \( \Delta \) is divisible by \( k \) (Option A).

Step 1: Understand Determinant Properties

The determinant \( \Delta \) of a 3x3 matrix is calculated using the rule of Sarrus or expansion by cofactors. Importantly, if all elements of any row (or column) in a matrix are divisible by a number \( k \), then the determinant of the matrix is also divisible by \( k \).

Step 2: Verify Divisibility Condition

Since each of the numbers \( A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}, A_{3} B_{3} C_{3} \) is divisible by \( k \), this implies that each row in the matrix has elements divisible by \( k \). Therefore, the determinant \( \Delta \) will be divisible by \( k \).

Step 3: Extend to Check for \( k^2 \)

For the determinant to be divisible by \( k^2 \), the matrix must have at least two rows or two columns with each element divisible by \( k \). In our case, though all rows are divisible by \( k \), this guarantees only that \( \Delta \) is divisible by \( k \), not \( k^2 \).

Step 4: Conclusion: Reason Solution

Thus, since each row is divisible by \( k \) but not necessarily forming a condition for \( k^2 \), \( \Delta \) is only divisible by \( k \) and not necessarily by \( k^2 \) or \( 2k \). The correct option is (A): \( \Delta \) is divisible by \( k \).

Key Concepts

Divisibility in Mathematics

Divisibility is an essential concept in mathematics, referring to how one integer can be evenly divided by another without leaving a remainder. In this context, the notion of divisibility applies to the numbers forming the elements of a matrix, which are stated to be divisible by a common integer, denoted as \( k \).

Understanding divisibility plays a crucial role when working with determinants of matrices. If each element in any row or column of a matrix is divisible by \( k \), the determinant of such a matrix is also divisible by \( k \). This property is powerful because it allows logical deduction about the product and relationships between numbers just by inspecting the divisibility.

• When a number \( a \) is divisible by another number \( b \), it can be expressed as a product of \( b \) and another integer. This forms the foundational principle behind mathematical operations involving divisibility.
• Knowing how to determine divisibility helps simplify complex mathematical computations and enhances problem-solving strategies.

Rule of Sarrus

The Rule of Sarrus is a simple method used to compute the determinant of a 3x3 matrix. Visualizing or understanding this rule involves writing out the matrix, duplicating the first two columns, and then calculating a series of products.

Specifically, for a 3x3 matrix \(\begin{pmatrix}A_1 & B_1 & C_1 \A_2 & B_2 & C_2 \A_3 & B_3 & C_3\end{pmatrix}\), the calculation step involves expanding the determinant along its diagonals:

• Take the products of the diagonals from top left to bottom right: \( A_1B_2C_3 + A_2B_3C_1 + A_3B_1C_2 \).
• Subtract the products of the diagonals from top right to bottom left: \( C_1B_2A_3 + C_2B_3A_1 + C_3B_1A_2 \).

The outcome after performing these operations provides the determinant of the matrix. Hence, this method simplifies the calculation process and is a great example of how structured techniques in mathematics can simplify what initially seems complex.

The simplicity of the Rule of Sarrus makes it particularly useful in academic settings for quickly verifying or computing determinants in 3x3 matrices.

Determinant Expansion by Cofactors

Determinant expansion by cofactors, also known as Laplace expansion, is another method used to evaluate the determinant of matrices. It involves expanding the determinant across a specific row or column by focusing on each element and its corresponding minor.

The minor of an element in a matrix is the determinant of the smaller matrix that remains after removing the element's row and column. For an element \( a_{ij} \), the cofactor is calculated as:

• Cofactor = \((-1)^{i+j}\) times the minor of the element.

To expand using cofactors in a 3x3 matrix \(\Delta = \begin{vmatrix}A_1 & B_1 & C_1 \A_2 & B_2 & C_2 \A_3 & B_3 & C_3\end{vmatrix}\), you can choose any row or column. For instance, expanding across the first row:

• The determinant \( \Delta \) is: \(A_1 \cdot \text{det}(\begin{vmatrix}B_2 & C_2 \B_3 & C_3\end{vmatrix}) - B_1 \cdot \text{det}(\begin{vmatrix}A_2 & C_2 \A_3 & C_3\end{vmatrix}) + C_1 \cdot \text{det}(\begin{vmatrix}A_2 & B_2 \A_3 & B_3\end{vmatrix})\)

This approach is flexible and can be adapted for larger matrices, making it valuable for solving determinant-related problems in various mathematical settings.