Questions
2 questions per paper
Difficulty
Medium
Importance
High yield for HPCL, NTPC, and ONGC
Overview
Linear Algebra is a fundamental pillar of engineering mathematics focused on vectors, matrices, and their transformations. For PSU exams like HPCL and NTPC, mastery here is crucial as it involves structured, calculation-based problems that are highly scoring. Students must focus on the properties of matrix operations and the conditions for existence of solutions in linear systems.
Matrix Algebra
Matrix algebra covers the basic arithmetic and advanced transformations of square and rectangular matrices. Exams frequently test properties related to determinants, rank, and inverse calculations.
- Determinant of triangular matrix is the product of diagonal elements
- Rank(A) <= min(m, n) for an m x n matrix
- A * adj(A) = det(A) * I
- A is invertible if and only if det(A) is not equal to zero
- Trace(A) = sum of diagonal elements = sum of eigenvalues
Systems of Linear Equations
This section deals with solving Ax = B using rank analysis and consistency conditions. Understanding the difference between unique, infinite, and no solutions is vital for quick elimination in MCQs.
- Consistent system: Rank(A) = Rank(A|B)
- Inconsistent system: Rank(A) is not equal to Rank(A|B)
- Infinite solutions exist if Rank(A) = Rank(A|B) < number of variables
- Homogeneous system (Ax=0) always has the trivial solution x=0
- Non-trivial solutions exist if det(A) = 0
Eigenvalues & Eigenvectors
Eigen-properties describe the behavior of linear operators and are a favorite topic for PSU examiners. You should be able to derive characteristic equations and identify properties of eigenvalues quickly.
- Characteristic equation: det(A - lambda * I) = 0
- Sum of eigenvalues = Trace(A)
- Product of eigenvalues = Determinant(A)
- Eigenvalues of A^k are lambda^k
- Eigenvalues of triangular/diagonal matrices are their diagonal elements
- Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation
Formula Sheet
A * adj(A) = det(A) * I
Rank(AB) <= min(Rank(A), Rank(B))
Sum of eigenvalues = sum of diagonal elements
Product of eigenvalues = det(A)
Ax = b => x = A^(-1)b
Cayley-Hamilton: P(lambda) = det(A - lambda*I) => P(A) = 0
Exam Tip
Use the property that the product of eigenvalues equals the determinant to eliminate incorrect options in seconds.
Common Mistakes
- Confusing the condition for trivial vs non-trivial solutions in homogeneous systems.
- Forgetting that row operations change the determinant (multiplication by a constant), but not the rank.
- Calculating eigenvalues from scratch instead of using trace and determinant properties to verify options.
More Revision Notes
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