Problem 63 - Algebra for JEE ADVANCED

Question

Value of \(\left|\begin{array}{ccc}1+x_{1} & 1+x_{1} x & 1+x_{1} x^{2} \\\ 1+x_{2} & 1+x_{2} x & 1+x_{2} x^{2} \\ 1+x_{3} & 1+x_{3} x & 1+x_{3} x^{2}\end{array}\right|\) depends upon a. \(x\) only b. \(x_{1}\) only c. \(x_{2}\) only d. none of these

Step-by-Step Solution

Answer

The correct answer is (d) none of these.

Step 1: Rewriting the Determinant

Let's denote the given determinant as \( D \) and write it as a 3x3 matrix: \[ D = \begin{vmatrix} 1+x_1 & 1+x_1x & 1+x_1x^2 \ 1+x_2 & 1+x_2x & 1+x_2x^2 \ \ 1+x_3 & 1+x_3x & 1+x_3x^2 \end{vmatrix} \]. This determinant is set up such that each row corresponds to terms related to \( x_1, x_2, x_3 \).

Step 2: Factor Common Terms from Columns

Observe that each entry in the first column is of the form \( 1 + x_i \), the second column is \( (1 + x_i)x \), and the third column is \( (1 + x_i)x^2 \). You can factor out the common factor \( 1 + x_i \) from each row. After factoring, the determinant becomes \( (1+x_1)(1+x_2)(1+x_3) \begin{vmatrix} 1 & x & x^2 \ 1 & x & x^2 \ 1 & x & x^2 \end{vmatrix} \).

Step 3: Evaluate the Simplified Determinant

Notice that the simplified determinant, \( \begin{vmatrix} 1 & x & x^2 \ 1 & x & x^2 \ 1 & x & x^2 \end{vmatrix} \), has identical rows. If any two rows in a determinant are identical, the determinant equals zero. Therefore, this simplified determinant is zero, implying \( D = (1+x_1)(1+x_2)(1+x_3) \cdot 0 = 0 \).

Step 4: Conclusion on Dependency

Since the determinant evaluates to zero regardless of the values of \( x, x_1, x_2, \) and \( x_3 \), the value of the determinant does not depend on \( x, x_1, x_2, \) or \( x_3 \). Hence, the determinant is constant and independent of any variable.

Key Concepts

Matrix Algebra

Matrix algebra forms the foundation for a wide range of mathematical computations, particularly when dealing with systems of linear equations. At its core, matrix algebra involves operations on matrices, such as addition, subtraction, and multiplication. These operations follow specific rules and properties that often mirror basic arithmetic but require careful adherence to matrix dimensions.

One interesting aspect of matrix algebra is the concept of the determinant, which is a scalar value that provides vital information about the matrix, such as whether it is invertible. For a square matrix, the determinant can help us determine whether a set of linear equations has a unique solution. In our exercise, we have a 3x3 matrix related to polynomials.

• Each entry of the matrix involves variables and their powers.
• Understanding the structure of the matrix is crucial for simplifying and computing the determinant effectively.

Overall, matrix algebra is essential for many fields, from engineering to computer graphics, because of its ability to simplify and solve complex problems involving large datasets and multiple variables.

Properties of Determinants

The determinant of a matrix is a key concept in linear algebra with several important properties. These properties often simplify the process of determining the overall behavior of the matrix.

### Factoring and Zero Rows A key property used in our exercise is the fact that the determinant of a matrix is zero if two rows or columns are identical. This property is crucial as it simplifies the determinant calculation:

• In our specific problem, factoring out common terms from each column led to rows that were exactly the same.
• As a result, the determinant was calculated to be zero.

### Other Important Properties
In addition to the zero determinant from identical rows, some other properties include:

• The determinant of an identity matrix is always 1.
• Swapping two rows (or columns) flips the sign of the determinant.
• Multiplying a row by a scalar multiplies the determinant by that same scalar.

Understanding these properties can drastically reduce the complexity of solving problems involving determinants, especially in larger matrices.

Linear Dependence

Linear dependence is an important concept when understanding linear algebra, particularly in relation to determinants. Linear dependence occurs when a set of vectors in a vector space can be expressed as a linear combination of others.

In the context of our matrix exercise, we can apply the idea of linear dependence to rows of a matrix. If rows are linearly dependent, they do not add any new information or dimension.

• This redundancy means the matrix does not span a full-dimensional space, contributing to the determinant being zero.
• In our exercise, this was demonstrated by observing that identical rows imply linear dependence, leading directly to a zero determinant.

### Visualizing Linear Dependence In simpler terms, visualize a scenario where two vectors point in the same or opposite directions. Adding more of those same vectors won't change the direction—they remain confined to a line. In a matrix, this leads to zero determinants and simplifies understanding matrix behavior. Thus, linear dependence is important in identifying the potential and limitations of sets of equations represented by matrices.