Questions
2 questions per paper
Difficulty
Medium
Importance
Medium yield for HPCL/NTPC/ONGC
Overview
Linear Algebra is a fundamental pillar of engineering mathematics involving the study of vectors, matrices, and linear systems. Mastering this topic is essential for PSU exams as it provides the algebraic backbone for solving complex network analysis, structural mechanics, and control system problems.
Matrices and Determinants
Matrices are rectangular arrays used to represent linear transformations, while determinants provide a scalar value that describes the scaling factor of these transformations. Understanding properties of determinants and matrix inverses is crucial for solving inverse problems quickly.
- A*adj(A) = det(A)*I
- det(A*B) = det(A)*det(B)
- Property of Singular Matrix: det(A) = 0
- Inverse exists only if matrix is non-singular
- Transpose Property: (AB)^T = B^T * A^T
Systems of Linear Equations
This subtopic focuses on determining the consistency of a system of equations using Rank analysis. Aspirants must differentiate between unique, infinite, and no-solution scenarios based on the relationship between the rank of the coefficient matrix and the augmented matrix.
- Consistent system: Rank(A) = Rank(A|B)
- Unique solution: Rank(A) = Rank(A|B) = Number of variables
- Infinite solutions: Rank(A) = Rank(A|B) < Number of variables
- Inconsistent system: Rank(A) is not equal to Rank(A|B)
- Cramer's rule is applicable only when det(A) is not 0
Eigenvalues and Eigenvectors
Eigenvalues represent the factor by which a vector is scaled during a linear transformation, while eigenvectors are the non-zero vectors that remain unchanged in direction. PSU exams frequently test properties of eigenvalues rather than demanding manual calculation of the characteristic equation.
- Sum of eigenvalues = Trace(A)
- Product of eigenvalues = det(A)
- Eigenvalues of A^k are λ^k
- Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation
- Eigenvalues of a triangular matrix are its diagonal elements
Formula Sheet
det(A*B) = det(A) * det(B)
A * adj(A) = |A| * I
Rank(A) = Rank(A^T)
Trace(A) = sum of diagonal elements = sum of eigenvalues
det(A) = product of eigenvalues
Characteristic Equation: det(A - λI) = 0
Rank(A) + Nullity(A) = n (number of variables)
Exam Tip
Always check if the matrix is triangular or singular before starting long calculations; PSU exams often hide simple shortcuts in these properties.
Common Mistakes
- Mistaking the sum of diagonal elements for something other than the trace or sum of eigenvalues.
- Forgetting that Row Echelon Form is not unique, which can confuse students during rank calculation.
- Attempting to calculate inverse matrices manually when properties of the adjugate matrix could solve the problem in seconds.
More Revision Notes
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