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Determinants - Cramer's Law Characteristic Polynomial | JEE 2025 | Namrata Ma'am

Vedantu JEE English・66 minutes read

The session focuses on solving systems of linear equations and characteristic polynomial equations, essential topics for JEE Main exams, emphasizing the importance of mastering Gram's rule and understanding determinants to determine solution uniqueness. It highlights the conditions for unique and non-unique solutions, as well as the practical application of matrix methods and characteristic polynomials to streamline problem-solving approaches.

Insights

  • The session highlights the significance of mastering systems of linear equations, particularly for JEE Main exams, as they consistently appear in the papers from 2019 to 2024 with a notable weightage of four marks, underscoring the necessity for students to be well-prepared in this area.
  • An upcoming series of videos focusing on previous year questions (PYQs) is set to launch, aimed at enhancing students' preparation strategies for JEE Main, with content scheduled for release in April and January, indicating a structured approach to exam readiness.
  • The concept of Gram's rule is introduced as an efficient method for solving systems of linear equations with three variables, offering an alternative to traditional methods like substitution or elimination, which can be more cumbersome and less systematic.
  • The text elaborates on the conditions for unique, no, or infinite solutions in linear equations, emphasizing the role of the primary determinant Delta, where a non-zero Delta indicates a unique solution, while a zero Delta necessitates further examination of other determinants to determine the nature of the solutions.
  • The discussion includes the characteristic polynomial and its relevance in linear algebra, explaining how to derive it from a square matrix, and emphasizes the practical importance of focusing on solving methods and algorithms over theoretical proofs, especially in exam contexts.

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Recent questions

  • What is a system of equations?

    A system of equations is a set of two or more equations with the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously. Systems can be classified as linear or non-linear, depending on the nature of the equations involved. In a linear system, each equation represents a straight line when graphed, and the solutions correspond to the points where these lines intersect. Solutions can be unique, infinitely many, or nonexistent, depending on the relationships between the equations. Understanding how to solve these systems is crucial in various fields, including mathematics, engineering, and economics.

  • How do you solve linear equations?

    Solving linear equations involves finding the values of the variables that make all equations true. Common methods include substitution, elimination, and using matrices. In substitution, one variable is expressed in terms of the others, and then substituted into the remaining equations. The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the others. For more complex systems, matrix methods can be employed, where the equations are represented in matrix form, and techniques like finding the inverse of the matrix or using determinants are applied. Each method has its advantages, and the choice often depends on the specific problem at hand.

  • What is a characteristic polynomial?

    A characteristic polynomial is a polynomial that is derived from a square matrix and is used to determine the eigenvalues of that matrix. It is calculated by taking the determinant of the matrix subtracted by a scalar multiple of the identity matrix, set to zero. For a matrix \( A \), the characteristic polynomial is expressed as \( \det(A - \lambda I) = 0 \), where \( \lambda \) represents the eigenvalues and \( I \) is the identity matrix. The roots of this polynomial give the eigenvalues, which are important in various applications, including stability analysis, quantum mechanics, and systems of differential equations. Understanding the characteristic polynomial is essential for analyzing the properties of matrices.

  • What is a homogeneous equation?

    A homogeneous equation is a type of linear equation where all constant terms are zero. This means that the equation can be expressed in the form \( Ax = 0 \), where \( A \) is a matrix and \( x \) is a vector of variables. Homogeneous systems always have at least one solution, known as the trivial solution, which is \( x = 0 \). However, they can also have infinitely many solutions if the determinant of the coefficient matrix is zero. The study of homogeneous equations is crucial in linear algebra, as it helps in understanding the structure of solutions and the behavior of linear transformations.

  • What is the Cayley-Hamilton theorem?

    The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. This means that if you have a matrix \( A \) and you compute its characteristic polynomial, substituting the matrix \( A \) into this polynomial will yield the zero matrix. This theorem is significant because it provides a way to express powers of matrices in terms of lower powers, facilitating computations in linear algebra. It also has applications in control theory, differential equations, and systems analysis, where understanding the behavior of matrices over time is essential. The theorem underscores the deep connection between linear algebra and polynomial algebra.

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Summary

00:00

Mastering Linear Equations for JEE Success

  • The session faced delays due to technical difficulties but has now commenced, focusing on two important topics: system of equations and characteristic polynomial equations, both relevant for JEE Main exams.
  • The first topic, system of linear equations, is crucial as it consistently appears in JEE Main papers from 2019 to 2024, with a weightage of four marks, making it essential for students to master.
  • An announcement was made regarding the upcoming launch of a previous year questions (PYQ) series, with videos scheduled for release in April and January attempts, aimed at helping students prepare effectively.
  • The session will cover the Gram's rule for solving systems of linear equations involving three variables (x, y, z) and three equations, emphasizing the efficiency of this method over traditional substitution or elimination techniques.
  • To apply Gram's rule, students must understand four determinants: Delta (primary determinant), Delta X, Delta Y, and Delta Z, which are formed using the coefficients of the equations and the constants from the equations.
  • The primary determinant Delta is constructed from the coefficients of x, y, and z from the three equations, while Delta X, Delta Y, and Delta Z are formed by replacing the respective columns with the constants from the equations.
  • The values of x, y, and z can be calculated using the formulas: x = Delta X / Delta, y = Delta Y / Delta, and z = Delta Z / Delta, providing a systematic approach to finding solutions.
  • The session explains the concepts of homogeneous and non-homogeneous systems, where a non-homogeneous system includes constant terms, affecting the nature of the solutions (unique, no solution, or infinitely many solutions).
  • The conditions for unique, no, or infinite solutions are outlined: a unique solution occurs when Delta is non-zero, while if Delta is zero, further checks on Delta X, Delta Y, and Delta Z determine if there are infinitely many solutions or no solution.
  • The terms consistent and inconsistent are defined, where a consistent system has at least one solution (either unique or infinite), while an inconsistent system has no solutions at all, highlighting the importance of understanding these concepts for solving equations effectively.

19:43

Linear Equations and Solution Conditions Explained

  • The text discusses a system of linear equations presented in a JEE Main question, identifying the equations as non-homogeneous due to the presence of non-zero constants (2, 5, and a + 1).
  • The primary determinant, Delta, is crucial for determining the nature of the solutions; if Delta is zero, the system may have infinite solutions or no solution, while a non-zero Delta indicates a unique solution.
  • The calculation of Delta involves the determinant of a 3x3 matrix formed by the coefficients of the equations, specifically Delta = 1, 1, 1; 2, 3, 2; 2, 3, a² - 1.
  • For Delta to equal zero, the condition a² - 1 = 2 must hold, leading to a² = 3, which gives a = ±√3, indicating that these values result in either infinite solutions or no solution.
  • The text emphasizes that for a unique solution, the value of 'a' must not equal ±√3, ensuring that Delta remains non-zero.
  • The next part of the discussion involves a different question where the conditions for infinite solutions are established, requiring all four determinants (Delta, Delta X, Delta Y, Delta Z) to be zero.
  • The calculation of Delta for the new question involves coefficients and constants, leading to Lambda = 3 and Mu = 7, which are derived from setting the appropriate determinants to zero.
  • The conditions for no solution are outlined, stating that Delta must be zero while at least one of the other determinants (Delta X, Delta Y, Delta Z) must be non-zero.
  • The text provides specific calculations for Lambda values (1 and -1/2) to determine the conditions for no solution, confirming that both values satisfy the criteria.
  • Finally, the conclusion states that the set of Lambda values resulting in no solution contains two elements: 1 and -1/2, confirming that both values meet the established conditions.

40:35

Homogeneous Equations and Non-Trivial Solutions

  • A homogeneous equation is defined as one where all constants are zero, meaning if D1, D2, and D3 are all zero, the equation is homogeneous, and the solution (0, 0, 0) will always satisfy it.
  • For homogeneous systems, there are only two possible outcomes: a unique solution or infinitely many solutions, with the unique solution being the trivial solution (0, 0, 0) when the determinant (Delta) is not equal to zero.
  • If Delta equals zero, the system has infinitely many solutions, referred to as non-trivial solutions, indicating that there are solutions other than the trivial one.
  • To find the number of values of theta for which a homogeneous system has a non-trivial solution, the determinant must be calculated and set to zero; this involves using the matrix formed by the coefficients of the equations.
  • The determinant D is calculated using the values from the equations, specifically: D = |137, 3Theta, 2; -1, 4, 7|, and must be solved for values of theta in the interval [0, π].
  • The equation derived from the determinant leads to a trigonometric equation that must be solved, specifically: 2cos(2Theta) = -3sin(3Theta).
  • The solutions for sin(Theta) are found to be sin(Theta) = 1/2, which corresponds to angles 30° (π/6) and 150° (5π/6) within the interval [0, π].
  • The final answer for the number of values of theta that satisfy the equation is two, as both angles are valid solutions within the specified interval.
  • A subsequent problem involves finding the value of K for which a system of linear equations has a non-trivial solution, requiring the determinant to be zero, leading to the equation 4K = -19, resulting in K = 9/2.
  • The final goal is to find the ratios x/y, y/z, and z/x from the equations, which involves manipulating the equations to eliminate variables systematically, ultimately leading to the desired ratios.

01:01:38

Solving Systems of Equations with Matrices

  • The text discusses a mathematical process involving the elimination of variables in a system of equations, specifically focusing on the relationships between variables X, Y, and Z, and how to manipulate these equations to find their ratios, such as Y/Z and X/Y, ultimately leading to a final answer of 1/2 for a specific problem.
  • The elimination of Y is described as a challenging step, requiring the multiplication of the equation by 3 to facilitate the process, resulting in a new equation that incorporates Z and simplifies the overall system.
  • The text emphasizes the importance of careful calculations, particularly when determining the ratios of the variables, with specific values derived from the equations, such as Y/Z being +1/2 and Z/X being -4, leading to the conclusion that X/Y equals -1/2.
  • A transition is made to discussing matrix methods for solving linear equations, highlighting the need to convert the system into matrix form, where Matrix A contains the coefficients of the variables and Matrix B contains the constants.
  • The determinant of Matrix A is calculated to determine the nature of the solutions; if the determinant is non-zero, a unique solution exists, which can be found by multiplying the inverse of Matrix A with Matrix B.
  • The process of finding the inverse of Matrix A is outlined, involving the calculation of the cofactor matrix and its adjoint, which is then used to derive the inverse necessary for solving the equations.
  • An example is provided where the determinant of Matrix A is calculated to be 6, indicating a unique solution, which is then found by multiplying the inverse of Matrix A with Matrix B, yielding the values of X, Y, and Z as 1, 2, and 3, respectively.
  • The text also covers homogeneous systems of equations, explaining that if the determinant is zero, it leads to either a trivial or non-trivial solution, with the condition for non-trivial solutions being that the determinant must equal zero.
  • The discussion includes the characteristic polynomial and equation, explaining that for any square matrix A, one can calculate the determinant of (A - λI) and set it equal to zero to find the characteristic polynomial, which will be quadratic for 2x2 matrices and cubic for 3x3 matrices.
  • The final part of the text suggests that while understanding the proofs behind these concepts is valuable, for practical purposes, especially in exam settings, focusing on the algorithms and methods for solving these problems is more efficient.

01:28:22

Finding Characteristic Polynomial of Matrices

  • The process of finding the characteristic polynomial of a matrix involves calculating \( A - xI = 0 \), where \( A \) is the matrix and \( I \) is the identity matrix. For a 3x3 matrix, this results in a cubic polynomial in \( x \) or \( \lambda \).
  • For the example given, the matrix \( A \) is defined as \( \begin{pmatrix} 1 & 2 & 0 \\ 2 & 1 & 2 \\ 3 & 0 & 3 \end{pmatrix} \). The identity matrix \( I \) is \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \), and the equation becomes \( A - xI = \begin{pmatrix} 1-x & 2 & 0 \\ 2 & 1-x & 2 \\ 3 & 0 & 3-x \end{pmatrix} \).
  • The determinant of \( A - xI \) must be calculated and set to zero to find the characteristic polynomial. The determinant simplifies to \( (2-x)(1-x)(3-x) - 0 = 0 \), leading to the cubic equation \( x^3 - 6x^2 + 7x + 2 = 0 \).
  • The value of \( K \) in the polynomial is determined by comparing coefficients, resulting in \( K = 2 \) after solving the equation.
  • The characteristic polynomial is defined as the polynomial obtained from \( \det(A - xI) \), while the characteristic equation is the polynomial set to zero. The left-hand side represents the characteristic polynomial, and the entire equation represents the characteristic equation.
  • The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. This means that once the characteristic polynomial is found, substituting \( x \) with the matrix \( A \) will yield a valid equation.
  • To find the inverse of a 2x2 matrix using the characteristic polynomial, the equation \( A - 6I = 10A^{-1} \) can be derived, leading to \( 10A^{-1} = A - 6I \).
  • A shortcut for finding the characteristic polynomial of a 2x2 matrix is given by the formula \( A^2 - \text{trace}(A)A + \det(A)I = 0 \), where the trace is the sum of the diagonal elements and the determinant is calculated from the matrix.
  • For a 3x3 matrix, the shortcut is \( A^3 - \text{trace}(A)A^2 + C A - \det(A)I = 0 \), where \( C \) is a constant derived from the matrix's properties. The trace and determinant must be calculated to apply this shortcut effectively.
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