Problem 86 - A Complete Resource Book in Mathematics for JEE Main

Question

5\. If \(\left|\begin{array}{ccc}0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0\end{array}\right|=0, a \neq b \neq c\), then (A) \(x=0\) if \(b(a+c) \leq a c\) (B) \(x=\) ? \(\sqrt{b(a+c)-a c}\) if \(b(a+c) \geq a c\) (C) \(x=0, \pm \sqrt{b(a+c)-a c}\) if \(b(a \neq c)>a c\) (D) None of these

Step-by-Step Solution

Answer

(C) is correct: If \(b(a+c) > a c\), \(x=0, \pm \sqrt{b(a+c)-a c}\).

Step 1: Identify Determinant Structure

Recognize that the determinant of a 3x3 matrix given is set to zero. The matrix is:\[\begin{bmatrix}0 & x-a & x-b \x+a & 0 & x-c \x+b & x+c & 0\end{bmatrix}\] We need to compute this determinant.

Step 2: Calculate the Determinant

Use the cofactor expansion along the first row to compute the determinant. The determinant is given by:\[0 \cdot (0 - (x-c)(x+c)) - (x-a) \cdot ((x+c)(x+b) - (x-c)(x)) + (x-b) \cdot ((x+a)(x-c) - (x-a)(x))\] Expanding and simplifying this expression should give us a polynomial equation in terms of \(x\).

Step 3: Simplify the Determinant Expression

Simplify the expression for the determinant by multiplying out and collecting like terms. The simplified polynomial should take the form of a quadratic equation:\[ f(x) = Ax^3 + Bx^2 + Cx \]Due to the structure of the matrix, the cube term is zero (determinant of skew-symmetric matrix of odd order is zero). Thus, it reduces to a quadratic equation.

Step 4: Set the Determinant to Zero and Solve

Set the simplified expression to zero to find values for \(x\):\[ Ax^2 + Bx + C = 0 \]The condition given is that this results in zero, which is a solvable quadratic. Consider the discriminant \(B^2 - 4AC\) of this quadratic to check the real values for \(x\).

Step 5: Analyze Solutions Based on Given Conditions

The determinant expression equates to zero, giving potential roots \(x = 0\) and \(x = \pm \sqrt{b(a+c) - ac}\). Evaluate the conditions based on \(b(a+c) - ac\) and rejections from original options. Thus,- If \(b(a+c) < ac\), the square root term inside is negative, thus only \(x = 0\) satisfies.- If \(b(a+c) \geq ac\), non-negative inside, thus real roots \(x = 0, \pm \sqrt{b(a+c) - ac}\) satisfy.

Key Concepts

3x3 Matrix

A 3x3 matrix is simply a square matrix that contains three rows and three columns. It is structured with nine elements, allowing it to represent complex multi-variable linear equations. A single 3x3 matrix can be used to solve systems of linear equations with three unknowns. This makes it an essential component in various fields, including physics, engineering, and computer graphics.

Key characteristics of a 3x3 matrix:

• Dimensions: 3 rows and 3 columns, often denoted as a grid of elements like \(\begin{bmatrix}a & b & c \ d & e & f \ g & h & i\end{bmatrix}\).
• Determinant: A numerical value computed from its elements, used in analysis and applications such as finding inverses and solutions to systems of equations.
• Across specialized contexts, a 3x3 matrix often enhances problem-solving strategies, particularly in linear algebra.

Understanding how elements within a 3x3 matrix interact through operations such as addition, multiplication, and finding determinants is crucial to mastering the broader concept of matrices.

Cofactor Expansion

Cofactor expansion, also known as Laplace’s expansion, is a method used to calculate the determinant of a matrix. For a 3x3 matrix, this involves choosing a row or a column to expand by calculating the determinant of smaller 2x2 matrices and multiplying by the elements of the chosen row or column.

The steps of cofactor expansion for a 3x3 matrix are:

• Select a row or column. Typically, the row or column with most zeros is chosen to simplify calculations.
• Multiply each element of that row or column by its cofactor. The cofactor is found by calculating the determinant of the smaller 2x2 matrix formed by removing the row and column of the element.
• Sum the products, applying a sign change pattern (+, -, +... or -, +, -...) based solely on the element's position.

For example, using the first row of a desired matrix, the cofactor expansion can provide a simplified expression for the determinant, which is crucial for solving equations involving matrices.

Quadratic Equation Solution

A quadratic equation is one that can be expressed in the form \(Ax^2 + Bx + C = 0\). Solving quadratic equations is a fundamental skill in algebra and involves finding the values of \(x\) that make the equation true.

Solving a quadratic equation typically involves the following methods:

• Factoring: Used when the quadratic can be written as a product of binomials.
• Completing the square: Manipulating the equation to form a perfect square trinomial.
• Quadratic formula: \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\), a robust solution that works for any quadratic equation.

The expression \(B^2 - 4AC\), known as the discriminant, indicates the nature of the roots. If this value is:

• Positive, there are two distinct real roots.
• Zero, there is one real root (a repeated root).
• Negative, there are no real roots (the roots are complex).

In the context of our problem, solving the determined quadratic coefficients helped reveal the values of \(x\) within a matrix determinant.

Skew-Symmetric Matrix

A skew-symmetric matrix is a special type of square matrix where the transpose of the matrix is equal to its negative. In mathematical terms, for a matrix \(A\), being skew-symmetric means \(A^T = -A\). An important property of skew-symmetric matrices is that they always have zeroes on the diagonal elements, making them look quite symmetrical despite their name.

Key properties:

• The determinant of an odd-order skew-symmetric matrix is always zero. This simplifies calculations significantly, especially when determining any matrix's invertibility or uniqueness.
• For a 3x3 skew-symmetric matrix, this determinant property means that the matrix cannot have a full rank, leading to a reduced set of possible solutions.
• They are common in physics, particularly in areas that involve rotational dynamics and systems like gyroscopes.

In the given exercise problem, by identifying the matrix as a skew-symmetric one, the determinant being zero confirmed the condition, enabling us to solve for specific values of \(x\) within the system.