Problem 18 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(a, b, c \in \mathbf{R}\) be all non-zero and satisfy \(a^{3}+b^{3}+c^{3}=2 .\) If the matrix \(A=\left(\begin{array}{lll}a & b & c \\\ b & c & a \\ c & a & b\end{array}\right)\) satisfies \(A^{T} A=I\), then a value of \(a b c\) can be : [Sep. \(\mathbf{0 2}, \mathbf{2 0 2 0}\) (II)] (a) \(-\frac{1}{3}\) (b) \(\frac{1}{3}\) (c) 3 (d) \(\frac{2}{3}\)

Step-by-Step Solution

Answer

The value of \(abc\) is \(\frac{2}{3}\).

Step 1: Understand the Problem

We need to find a specific value for \(abc\) given certain conditions. The matrix \(A\) is orthogonal since \(A^T A = I\), and the cube sum equation gives us \(a^3 + b^3 + c^3 = 2\). We'll use both conditions to find an expression for \(abc\).

Step 2: Utilize Orthogonality of the Matrix

Since \(A^T A = I\), the rows of \(A\) are orthonormal. This implies their dot products with each other give 0. Calculate the dot products of different pairs of rows of \(A\) and set them equal to 0.

Step 3: Explore Dot Products

Compute the dot products: 1. First and second rows: \(ab + bc + ca = 0\).2. First and third rows: \(ac + ab + bc = 0\).3. Second and third rows: \(bc + ac + ab = 0\).These conditions are consistent with each other and simplify to \(ab + bc + ca = 0\).

Step 4: Analyze Given Expression

From the given expression \(a^3 + b^3 + c^3 = 2\), and knowing that \(ab + bc + ca = 0\), we can use the identity \[a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)\] which simplifies to \(a^3 + b^3 + c^3 = 3abc\) under the conditions given because \(ab + bc + ca = 0\).

Step 5: Substitute Values and Solve

From Step 4, we have \(a^3 + b^3 + c^3 = 3abc = 2\). Thus, \(abc = \frac{2}{3}\).

Step 6: Verify Solutions against Choices

Among the given options, \(abc = \frac{2}{3}\) corresponds with option (d).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is an important operation in vector mathematics. It is defined for two vectors of equal dimensions and results in a scalar quantity. If we have vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product is given by:\[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \cdots + u_n v_n \]In simpler terms, you multiply the corresponding pairs of vector elements and sum them all up.In this exercise, the dot product is utilized to show the orthogonality of the rows in matrix \( A \). Given that \( A^T A = I \), where \( I \) is the identity matrix, it implies the rows of the matrix are orthonormal.

• The dot product between different pairs of rows, such as the first and second rows, should result in zero: \( ab + bc + ca = 0 \).
• This concept is fundamental in obtaining the values required by the problem, thereby highlighting the importance of dot products in verifying orthogonality.

Orthogonal Matrix

An orthogonal matrix is a square matrix whose rows and columns are orthogonal unit vectors. This means that the matrix satisfies the condition \( A^T A = I \), where \( I \) is the identity matrix.
Orthogonal matrices are integral in various mathematical computations due to:

• Their property of conserving the length of vectors after transformation.
• Being easily invertible, with the inverse equal to the transpose, \( A^{-1} = A^T \).

In the exercise at hand, recognizing \( A \) as an orthogonal matrix allowed us to interpret that its rows are orthogonal. The orthogonality condition helped in simplifying the relationships between \( a, b, \) and \( c \) using dot products, ultimately leading to determining the value of \( abc \).The structural nature of \( A \), given the non-zero vectors \( a, b, c \), showcases the elegance of orthogonal matrices simplifying complex problems such as these.

Cubic Equation

A cubic equation is a polynomial equation of degree three. It typically takes the form:\[ ax^3 + bx^2 + cx + d = 0 \]Cubic equations can have one real root and two complex conjugate roots, or three real roots. They involve more complexity than quadratic equations due to:

• The fact that their solutions cannot be found using a simple general formula like the quadratic formula.
• Requiring more elaborate algebraic methods or numerical solutions.

In our exercise, the cubic expression \( a^3 + b^3 + c^3 = 2 \) is evaluated. Combined with the identity \( a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \) and the special condition \( ab + bc + ca = 0 \), this equation simplifies to precisely determine \( abc \).Understanding cubic equations and their relationships, such as these simplifications in terms of symmetrical identities, can significantly aid in breaking down and solving intricate problems. The cubic equation in this case helped us find that the sum leads to the conclusion \( abc = \frac{2}{3} \), matching the provided solution.