Questions
2 questions in typical PSU papers
Difficulty
Medium
Importance
High yield for GATE/PSU screening
Overview
Linear Algebra is a foundational branch of mathematics focusing on vectors, matrices, and linear transformations. In PSU exams, it is highly valued for its predictable numerical nature, allowing candidates to score quickly by applying standardized algorithms for matrix operations and eigenvalue computation.
Matrices & Determinants
Matrices are rectangular arrays used to represent linear maps and systems of equations. Mastery involves understanding algebraic operations, properties of determinants, and specific matrix types like symmetric, skew-symmetric, and orthogonal matrices.
- Det(AB) = Det(A) * Det(B)
- A * adj(A) = |A| * I
- Inverse exists only if |A| is non-zero
- For orthogonal matrix A, A * A^T = I
- Trace(A) = sum of diagonal elements
Systems of Linear Equations
This subtopic focuses on solving systems Ax=b using rank-based analysis. Candidates must distinguish between consistent (unique or infinite solutions) and inconsistent systems based on the Rank-Nullity Theorem and augmented matrix properties.
- Consistent if Rank(A) = Rank(A|B)
- Unique solution if Rank(A) = number of variables
- Infinite solutions if Rank(A) < number of variables
- Homogeneous system AX=0 always has a trivial solution X=0
- Cramer's Rule is applicable only for square systems with non-zero determinant
Eigenvalues & Eigenvectors
Eigenvalues are scalars that scale eigenvectors during a linear transformation. These are essential for diagonalizing matrices and solving differential equations commonly appearing in PSU technical tests.
- Characteristic Equation: |A - λI| = 0
- Sum of eigenvalues = Trace(A)
- Product of eigenvalues = |A|
- Eigenvalues of A^k are λ^k
- Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation
Formula Sheet
|A| = sum of λ_i
Trace(A) = sum of diagonal elements
Rank(A|B) = Rank(A) for consistency
A^-1 = (1/|A|) * adj(A)
det(kA) = k^n * det(A) for n x n matrix
Eigenvalues of triangular/diagonal matrix are diagonal elements
Exam Tip
Always use the Cayley-Hamilton theorem to find inverse matrices or high powers of a matrix, as it is significantly faster than standard row reduction.
Common Mistakes
- Assuming the rank of a product AB is the same as rank A or rank B, rather than calculating it using Rank(AB) <= min(Rank A, Rank B).
- Forgetting to check the consistency condition (Rank A = Rank A|B) before attempting to solve for variables.
- Miscalculating the determinant for matrices larger than 3x3, often ignoring row/column reduction properties.
More Revision Notes
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