Problem 9 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a_{i}, b_{i}, c_{i} \in R(i=1,2,3)\) and \(x \in R\) and \(\left|\begin{array}{lll}a_{1}+b_{1} x & a_{1} x+b_{1} & c_{1} \\ a_{2}+b_{2} x & a_{2} x+b_{2} & c_{2} \\ a_{3}+b_{3} x & a_{3} x+b_{3} & c_{3}\end{array}\right|=0\), then (A) \(\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\\ a_{3} & b_{3} & c_{3}\end{array}\right|=4\) (B) \(x=\pm 1\) (C) \(x=2\) (D) None of these

Step-by-Step Solution

Answer

Option B: x = ±1.

Step 1: Understand the Matrix Determinant Condition

Given the matrix determinant must equal zero, recognize that the determinant of a matrix represents a linear transformation's scaling factor. A determinant of zero indicates the transformation is not invertible, meaning there is no unique solution, suggesting dependence between rows.

Step 2: Simplify and Analyze the Matrix

Rewrite the determinant condition \[\left|\begin{array}{lll}a_{1}+b_{1} x & a_{1} x+b_{1} & c_{1} \ a_{2}+b_{2} x & a_{2} x+b_{2} & c_{2} \ a_{3}+b_{3} x & a_{3} x+b_{3} & c_{3}\end{array}\right| = 0\]and analyze how each row transforms. Notice that a determinant of zero often arises from rows being linearly dependent or some specific value of \(x\) that satisfies the condition.

Step 3: Consider the Special Case of Linear Dependence

Recognize that linear dependence between the rows of the matrix can occur for special values of \(x\). Consider the potential these transformations lead to dependencies that force the determinant to zero when extended to specific \(x\) values.

Step 4: Check for Specific Values of x

To see if \(x\) influences the determinant beyond structural dependence, evaluate how plugging specific typical values of \(x\) like \(x = 1\) or \(x = -1\) might reduce the matrix to identity or scalar multiples, effectively zeroing the determinant across certain conditions.

Step 5: Verify x using Simplicity Analysis

By setting \(x = 1\) and \(x = -1\) based on typical scenarios and re-solving for the determinants, infer if any x-values consistently result in singular, non-invertible matrices. If found consistently and with no contrary evidence, ratify them as solutions.

Key Concepts

Linear Dependence

Linear dependence is a fundamental concept in the study of vectors and matrices. When we talk about linear dependence, we refer to a situation where one vector can be expressed as a linear combination of others in the set. In the context of a matrix, it means that the rows or columns are not linearly independent from each other.
In simpler terms, if one row or column in a matrix can be made by adding, subtracting, or scaling the other rows or columns, they are said to be linearly dependent. This dependency signals redundancy in information. It implies that the vectors are not mailing the fullest pattern of directions available in the space.
Key points of linear dependence include:

• A determinant of zero often signifies linear dependence among rows or columns.
• If vectors (or rows/columns) are linearly dependent, they cannot span a full-dimensional space.
• Linear dependence implies there is no unique solution when solving systems of equations, as dependencies prevent invertibility.

In our exercise, recognizing linear dependence is crucial when we see the determinant equaling zero. It helps us understand that the rows of the matrix might be aligning in such a way that they cancel each other's effect, leading to zero scaling on transformation.

Linear Transformation

A linear transformation in mathematics is a function that maps vectors to other vectors in a way that preserves vector addition and scalar multiplication. In matrix terms, a linear transformation can be represented by the operation done by multiplying a vector by a matrix.
The role of a determinant is significant here. The determinant of a matrix tells us about the scaling factor of the transformation. If it's zero, the transformation squashes the space into a lower dimension, indicating the transformation might be degenerative, projecting all inputs into a confined subspace.

• The determinant's absolute value offers insight into how much expansion or contraction occurs during the transformation.
• A negative determinant indicates the transformation results in a reflection over a certain axis in the vector space.

Linear transformations have applications in various fields, including computer graphics, where they help in tasks like rotation, scaling, and reflection of images in graphics rendering. In the given exercise, understanding that a zero determinant denotes the loss of dimensions in space transformation is essential.

Invertible Matrix

An invertible matrix is one that can be "reversed" or "undone" by another matrix, known as its inverse. If you multiply a matrix by its inverse, you will get the identity matrix, which acts as a sort of "1" for matrices.
A key property of invertible matrices is that they must have a non-zero determinant. A zero determinant means the matrix cannot be inverted because some information is missing or compressed into fewer dimensions.

• An invertible matrix has full rank, meaning all its rows or columns are linearly independent.
• Invertible matrices ensure a unique solution for systems of linear equations.
• If a matrix is singular (non-invertible), it indicates at least one linear dependence, evident as a zero determinant.

Understanding invertibility is crucial in applications like solving linear system equations, where having an invertible coefficient matrix assures that the system has a unique solution. In our exercise, a determinant of zero points to the failure in invertibility because the transformation cannot be uniquely reversed, highlighting linear dependence within the rows or columns.