Questions
2 questions per paper
Difficulty
Medium
Importance
High yield for HPCL/NTPC/ONGC
Overview
Linear Algebra is a fundamental pillar of engineering mathematics, focusing on the study of vectors, matrices, and linear transformations. For PSU exams like HPCL and ONGC, mastering matrix operations and eigenvalue theory is essential as these concepts appear frequently in numerical analysis and system modeling.
Matrices & Determinants
Matrices are the primary tool for representing linear systems and transformations. Understanding properties of determinants and special matrices is crucial for identifying singular systems and performing efficient calculations.
- Determinant of a triangular matrix is the product of its diagonal elements.
- A matrix is singular if its determinant is zero.
- Inverse exists if and only if determinant is non-zero (A inverse = adj(A)/det(A)).
- Trace of a matrix is the sum of its diagonal elements.
- Orthogonal matrix: A transpose multiplied by A equals Identity matrix.
Systems of Linear Equations
Solving systems involves analyzing the rank of the augmented matrix to determine the nature of the solution set. PSU exams often test the conditions for consistency using the Rank-Nullity theorem.
- Consistent system: Rank(A) = Rank(A|B).
- Unique solution if Rank(A) = Rank(A|B) = Number of variables.
- Infinite solutions if Rank(A) = Rank(A|B) < Number of variables.
- No solution if Rank(A) is not equal to Rank(A|B).
- Homogeneous system AX=0 always has at least the trivial solution.
LU Decomposition
LU decomposition breaks a square matrix into a lower triangular matrix and an upper triangular matrix, simplifying the process of solving linear systems. This is frequently used for computational efficiency in engineering algorithms.
- A = LU where L is unit lower triangular and U is upper triangular.
- Used to solve AX = b by solving Ly = b then Ux = y.
- Particularly efficient for solving multiple systems with the same coefficient matrix.
- Requires row operations during decomposition for non-singular matrices.
Eigenvalues & Eigenvectors
Eigenvalues represent the scaling factors of linear transformations, while eigenvectors denote the directions preserved under that transformation. These concepts are frequently tested through characteristic equations and the Cayley-Hamilton theorem.
- Characteristic equation: det(A - lambda*I) = 0.
- Sum of eigenvalues equals the Trace of the matrix.
- Product of eigenvalues equals the Determinant of the matrix.
- Cayley-Hamilton theorem: Every square matrix satisfies its own characteristic equation.
- Eigenvalues of A transpose are the same as A.
Formula Sheet
det(A) = product of eigenvalues
trace(A) = sum of eigenvalues
A*adj(A) = det(A)*I
Rank(A+B) <= Rank(A) + Rank(B)
A*A^(-1) = I
Characteristic Equation: |A - λI| = 0
Exam Tip
Always verify if a matrix is singular using the determinant before attempting to find the inverse or solving the linear system.
Common Mistakes
- Confusing the property of the sum of eigenvalues equaling the trace with the sum equaling the determinant.
- Forgetting to check the consistency condition (Rank equality) before attempting to solve a system.
- Neglecting the trivial solution in homogeneous systems where the determinant is zero.
More Revision Notes
Ready to test yourself?
Play topic-wise Linear Algebra questions in Aspirant Arcade — gamified MCQ practice.
Download Free