Problem 45 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(D=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|\) for \(x \neq 0, y \neq 0\), then D is (a) divisible by \(x\) but not \(y\) (b) divisible by \(y\) but \(\operatorname{not} x\) (c) divisible by neither \(x\) nor \(y\) (d) divisible by both \(x\) and \(y\)

Step-by-Step Solution

Answer

D is divisible by both x and y; option (d).

Step 1: Understand the Problem

We are given a 3x3 determinant matrix \( D \) and need to determine its divisibility by \( x \) and \( y \). The matrix provided is \( D = \left| \begin{array}{ccc} 1 & 1 & 1 \ 1 & 1+x & 1 \ 1 & 1 & 1+y \end{array} \right| \). We are tasked to figure out whether \( D \) is divisible by \( x \), \( y \), both, or neither.

Step 2: Use Basic Determinant Properties

Recall that a determinant can often be simplified using row operations that do not alter its value. Subtract the first row from the other rows to introduce zeros, potentially simplifying the calculation. For this matrix, perform the operation on the second and third rows: \[ R_2 = R_2 - R_1, \quad R_3 = R_3 - R_1 \].

Step 3: Simplify the Matrix

Apply the operations. The new matrix becomes: \[ \left| \begin{array}{ccc} 1 & 1 & 1 \ 0 & x & 0 \ 0 & 0 & y \end{array} \right| \]. Notice that you've created a matrix with zeros that simplify the determinant calculation.

Step 4: Calculate the Determinant

For a diagonal matrix of the form \( \left| \begin{array}{ccc} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{array} \right| \), the determinant is the product \( abc \). So for our matrix, the determinant is \( D = 1 \cdot x \cdot y = xy \).

Step 5: Determine the Divisibility of D

The determinant \( D = xy \) is clearly divisible by both \( x \) and \( y \), as \( x \) and \( y \) are factors in the product. This means that the answer is option (d): \( D \) is divisible by both \( x \) and \( y \).

Key Concepts

Matrix Operations

Matrix operations play a crucial role in solving problems related to determinants, especially in linear algebra. Understanding basic matrix operations like row operations can greatly simplify complex calculations. When dealing with determinants, we often use transformations such as row or column operations to introduce zeros in a matrix.
These operations help simplify the determinant calculation without changing its value. In our exercise, subtracting the first row from the second and third rows of the given 3x3 matrix helps reduce it into a simpler form. This eliminated terms and allowed easy determination of the determinant.

• Row operations include swapping, multiplying a row by a non-zero scalar, and adding or subtracting rows.
• These do not alter the value of a determinant when applied correctly.

Understanding and utilizing these operations can make working with matrices much more efficient and intuitive.

Divisibility

Divisibility in mathematics typically refers to the ability of one number to be evenly divided by another without leaving a remainder. In the context of determinants, we examine if the determinant can be expressed in terms of its variables.
In the given exercise, the transformed matrix presents the determinant as a product of terms including the variables \(x\) and \(y\). Calculating a determinant that results in the expression \(xy\) shows that it is divisible by both \(x\) and \(y\).

• If a determinant can be expressed with its components being part of a product, it suggests divisibility by each factor of the product.
• Analyzing the structure and values of a matrix is key to determining such divisibility.

In simple terms, for a product like \(xy\), both \(x\) and \(y\) are integral divisors, confirming the matrix’s determinant divisibility by these variables.

3x3 Matrix

A 3x3 matrix is a fundamental type of square matrix used extensively in linear algebra. It consists of three rows and three columns, and determinants of such matrices provide significant insights into their properties.
Determinants help determine the invertibility of the matrix, among other attributes. In this problem, we work on a 3x3 matrix to understand its determinant.

• 3x3 matrices can represent systems of linear equations, transformations, or other mathematical models.
• The calculation of a 3x3 matrix determinant is performed by using both cofactor expansion and row reduction techniques for simplification.

The matrix operations applied here reduced it to a diagonal form, simplifying our work to find the determinant, which in turn assisted in assessing its divisibility properties.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It involves matrices, systems of linear equations, and their transformations.
In linear algebra, the determinant is a vital concept that aids in solving systems of equations, finding volumes of geometric bodies, and more. The exercise tackled here is a great example of applying linear algebra principles to determine matrix properties.

• Determinants help ascertain whether a matrix is invertible, as a non-zero determinant confirms invertibility.
• They play a role in calculating eigenvalues and eigenvectors, further broadening their application scope.

Mastering these core ideas in linear algebra, such as handling determinants of matrices, can greatly ease understanding more complex mathematical concepts and practical problems.