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

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 \(\Delta=\left|\begin{array}{lll}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|\) 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 \(k^2\), so the correct answer is (B).

Step 1: Understanding the Determinant

The given determinant is a 3x3 matrix:\[\Delta=\left|\begin{array}{lll}A_{1} & B_{1} & C_{1} \A_{2} & B_{2} & C_{2} \A_{3} & B_{3} & C_{3}\end{array}\right|\]We need to determine the divisibility properties of this determinant with respect to the integer \(k\), by using the property that each row is divisible by \(k\). This means each element of the determinant is divisible by \(k\).

Step 2: Property of Matrix Determinants

A property of determinants states that if all elements of a row (or a column) of a matrix are divisible by a constant \(k\), then the determinant is divisible by that constant \(k\). In this case, all rows are divisible by \(k\), implying the determinant is divisible by \(k\).

Step 3: Applying the Property to Multiple Rows

Because each of the three rows is divisible by \(k\), the determinant \(\Delta\) is not only divisible by \(k\) for one row, but it is also highly likely to be further divisible due to shared divisibility across multiple rows.

Step 4: Conclusion on Divisibility

If every row is divisible by \(k\), the determinant becomes divisible by \(k^3\), because for every row added, divisibility adds another factor of \(k\). So \(\Delta\) is divisible by \(k^2\) and possibly even higher powers of \(k\). Nonetheless, because the basic property calculated up to \(k^2\) provides significant confirmation, option (B) is correct.

Key Concepts

Matrix Properties

In linear algebra, matrices come with distinct sets of properties that help simplify complex operations. A matrix is essentially a rectangular array of numbers arranged in rows and columns. The determinant of a matrix, as encountered in this exercise, is a unique value that can be derived from its elements. The determinant has several key properties:

• Linearity: The determinant is a linear function concerning each of the matrix's rows and columns. This means you can add and multiply rows (or columns) to influence the determinant's value while following specific rules.
• Determinant of Identity Matrix: The determinant of an identity matrix (where all diagonal elements are 1 and all other elements are 0) is always 1. This property allows easy verification of basic matrix operations.
• Effect of Row Operations: Multiplying a row by a scalar multiplies the determinant by that scalar. Similarly, adding a multiple of one row to another doesn't change the determinant.

Understanding these properties allows you to assess problems involving determinants, like verifying their divisibility or solving systems of equations.

Divisibility

Divisibility refers to the ability of one integer to be divided by another without leaving a remainder. In this exercise, we explored how a determinant relates to divisibility. When the elements of a matrix's rows (or columns) are all divisible by a particular integer, it significantly impacts the divisibility of the determinant itself.

• Row Divisibility Impact: If every element in a row of a matrix is divisible by some integer \(k\), then the determinant is divisible by \(k\). This is due to the linearity property of determinants.
• Multiple Rows and Higher Powers: When all rows (or columns) of a matrix are divisible by the same integer \(k\), the determinant becomes divisible by higher powers of \(k\). Specifically, if each row is divisible by \(k\), the determinant will at minimum be divisible by \(k^3\) for a 3x3 matrix, due to the cubic nature of the determinant calculation.

These principles of divisibility allow for efficient problem-solving strategies in linear algebra and related mathematical fields.

Linear Algebra

Linear algebra is a branch of mathematics focused on vector spaces and linear mappings between them. It includes the study of lines, planes, and subspaces, but is especially significant for understanding matrices and their determinants. Here are a few key aspects of linear algebra crucial for this exercise:

• Determinant Calculation: Determinants are an essential aspect of matrix theory, used to determine the invertibility of a matrix, solve linear systems, and assess geometric properties like volume in higher dimensions.
• Matrix Operations: Linear algebra makes use of operations such as matrix addition, subtraction, and multiplication. The rules governing these operations are foundational, as they lead to properties like the distributive law and the determination of matrix inverses.
• Applications: This branch of mathematics is widely applicable in various fields such as physics, computer science, and engineering. It aids in solving systems of linear equations, performing transformations, and even creating algorithms for computer applications.

By understanding the key concepts and operations of linear algebra, students can tackle problems involving determinants and their properties with greater confidence.