Problem 40 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(a, b, c\), are non zero complex numbers satisfying \(a^{2}+b^{2}+c^{2}=0\) and \(\left|\begin{array}{ccc}b^{2}+c^{2} & a b & a c \\ a b & c^{2}+a^{2} & b c \\\ a c & b c & a^{2}+b^{2}\end{array}\right|=k a^{2} b^{2} c^{2}\), then \(k\) is equal to [Online May 19, 2012] (a) 1 (b) 3 (c) 4 (d) 2

Step-by-Step Solution

Answer

k = 2.

Step 1: Understand the Problem Statement

We need to solve a matrix determinant problem with complex numbers satisfying a given condition. According to the problem, the sum of squares of three complex numbers \( a, b, c \) equals zero, i.e., \( a^2 + b^2 + c^2 = 0 \). We're tasked with finding the value of the constant \( k \) such that the determinant of a given 3x3 matrix equals \( k a^2 b^2 c^2 \).

Step 2: Substitute into the Matrix

Let's rewrite the given matrix:\[ M = \begin{pmatrix} b^2 + c^2 & ab & ac \ ab & c^2 + a^2 & bc \ ac & bc & a^2 + b^2 \end{pmatrix} \] We are to compute the determinant of this matrix.

Step 3: Expand the Determinant

Compute the determinant of matrix \( M \) using the standard determinant expansion formula for a 3x3 matrix. \[ \text{det}(M) = (b^2 + c^2)((c^2 + a^2)(a^2 + b^2) - (bc)^2) - ab(ab(a^2 + b^2) - ac(bc)) + ac(ab(bc) - ac(c^2 + a^2)) \] Simplify each term to find the determinant formula.

Step 4: Simplify Using Given Condition

Given that \( a^2 + b^2 + c^2 = 0 \), simplify the expression. Substitute \( a^2 + b^2 = -c^2 \), \( b^2 + c^2 = -a^2 \), and \( c^2 + a^2 = -b^2 \) into the expression.This simplification helps to cancel out several terms. Follow through the algebraic manipulation accordingly.

Step 5: Calculate the Determinant

After substitution and simplification, the determinant equates to \ 2 a^2 b^2 c^2. Thus, \( \text{det}(M) = 2 a^2 b^2 c^2 \).

Step 6: Compare with \( k a^2 b^2 c^2 \)

According to the problem, \( \text{det}(M) = k a^2 b^2 c^2 \). Since we've calculated \( \text{det}(M) = 2 a^2 b^2 c^2 \), it follows that \( k = 2 \).

Key Concepts

Matrix Determinants

The concept of matrix determinants is vital in linear algebra and many applied mathematics domains. In simple terms, a determinant is a special number that can be calculated from a square matrix. It provides key insights into the matrix's properties.When we have a 3x3 matrix, the determinant is computed using a specific formula that involves the matrix's elements. For example, given a matrix \( M = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the determinant can be found as:\[\text{det}(M) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula shows that each element of the matrix is multiplied by the determinant of the 2x2 submatrix that remains after removing its row and column. It's a way to "sum up" the matrix's behavior with respect to linear transformations.For complex matrices, like in the original exercise, the same principles apply, but you must consider the properties of complex numbers when evaluating each element's contribution.

Complex Algebra

Complex algebra involves numbers written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \). Complex algebra allows us to perform arithmetic operations such as addition, subtraction, multiplication, and division on these numbers.Understanding complex numbers is essential when encountering matrices that include them, as in our exercise. To resolve these, you add and subtract as you would with real numbers, applying the underlying rule that \( i^2 = -1 \).In the original problem, complex numbers \( a, b, c \) are used in the matrix determinants. Key to the solution was the condition \( a^2 + b^2 + c^2 = 0 \), which indicates specific relationships among these numbers. This seems peculiar in real numbers, but with complex numbers, it can mean the imaginary parts are canceling out to zero. It's crucial to explore these properties to simplify the determinant calculation and find the value of \( k \).

JEE Mathematics

The JEE Mathematics syllabus is an extensive program that includes complex numbers and determinants of matrices, as seen in our example exercise. It's designed to test students' understanding and ability to apply mathematical concepts in practical contexts.The problem at hand is a typical type of question one might find in JEE exams. It requires not only the ability to perform basic calculations but also to apply algebraic identities creatively. Students often must deduce relationships between variables based on given conditions, like using the information \( a^2 + b^2 + c^2 = 0 \) to simplify the determination of the matrix's determinant.Key strategies to tackle these types of JEE questions include:- Identifying relevant mathematical ideas.- Simplifying complex expressions using given conditions or properties.- Applying knowledge from various topics, such as complex algebra, to reach the correct solution efficiently.Developing confidence and competence in these areas can greatly help students in JEE Mathematics, providing a strong foundation for further studies.