Questions
1–2 questions in most PSU papers
Difficulty
Medium
Importance
High yield for HPCL/NTPC technical sections
Overview
Linear Algebra is a foundational branch of mathematics focusing on vectors, matrices, and systems of linear equations. It is essential for PSU exams as it provides the mathematical framework for solving complex problems in engineering disciplines like electrical and mechanical systems. Aspirants must master matrix operations and eigendecomposition to quickly solve numerical problems.
Matrix Algebra
Matrix algebra involves fundamental operations on square and rectangular matrices, including transpose, inverse, and determinant calculations. Exams often test properties of special matrices like symmetric, skew-symmetric, orthogonal, and unitary matrices.
- A transpose is symmetric if A = A^T and skew-symmetric if A = -A^T
- A matrix is orthogonal if A * A^T = I
- Determinant of a product is the product of determinants: |AB| = |A| * |B|
- Inverse exists only if the matrix is non-singular, i.e., |A| != 0
- Trace(A) = sum of diagonal elements = sum of eigenvalues
Systems of Linear Equations
This sub-topic focuses on finding solutions for AX = B using matrix methods and Rank analysis. The consistency of the system is determined by the Rank of the coefficient matrix versus the Rank of the augmented matrix [A|B].
- Consistent system: Rank(A) = Rank(A|B)
- Unique solution: Rank(A) = Rank(A|B) = number of unknowns (n)
- Infinite solutions: Rank(A) = Rank(A|B) < n
- Inconsistent system: Rank(A) != Rank(A|B)
- Cramer's rule is only applicable for square systems where |A| != 0
Eigenvalues and Eigenvectors
Eigenvalues are roots of the characteristic equation |A - λI| = 0. They represent fundamental scalar values that describe the transformation properties of the matrix and are heavily tested in PSU technical papers.
- Characteristic equation: det(A - λI) = 0
- Sum of eigenvalues = Trace(A)
- Product of eigenvalues = det(A)
- Eigenvalues of A^k are λ^k
- Eigenvalues of A^-1 are 1/λ
- Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation
Formula Sheet
A * A^-1 = I
|A - λI| = 0
Rank(A) <= min(m, n)
AX = B => X = A^-1 * B
Sum(λ_i) = Σ a_ii
Product(λ_i) = |A|
A^n - c1*A^(n-1) + ... + (-1)^n * |A| * I = 0 (Cayley-Hamilton)
Exam Tip
Always use the Trace and Determinant properties to verify your eigenvalues before wasting time performing full polynomial expansion.
Common Mistakes
- Confusing the sum of eigenvalues with the determinant instead of the trace
- Forgetting to check the consistency criteria (Rank) before attempting to solve for variables
- Miscalculating the determinant for matrices larger than 3x3 during high-pressure exam conditions
More Revision Notes
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