Questions
2 questions per paper
Difficulty
Medium
Importance
Medium yield for BHEL/NTPC/ONGC
Overview
Linear Algebra is a fundamental branch of mathematics focusing on matrices, vector spaces, and linear transformations. In PSU exams, it serves as a high-scoring area because the problems are standard, procedural, and heavily reliant on properties rather than complex derivations.
Matrix Algebra
Matrix operations form the foundation of this module, covering determinants, inverses, and special matrix types. Exam questions frequently test your knowledge of properties of determinants and the behavior of square matrices.
- Determinant of A*B = Det(A) * Det(B)
- A matrix is singular if Det(A) = 0
- Inverse exists only if the matrix is non-singular
- Trace(A) = sum of diagonal elements
- Orthogonal matrix condition: A * A^T = I
- Symmetric matrix: A = A^T; Skew-symmetric: A = -A^T
Systems of Linear Equations
This section involves analyzing whether a system of equations AX = B has a unique solution, infinite solutions, or no solution. The rank of the matrix relative to the augmented matrix is the decisive factor for consistency.
- Consistent system: Rank(A) = Rank(A|B)
- Unique solution: Rank(A) = Rank(A|B) = Number of unknowns
- Infinite solutions: Rank(A) = Rank(A|B) < Number of unknowns
- Inconsistent system: Rank(A) != Rank(A|B)
- Trivial solution exists if Rank(A) = n for AX = 0
Eigenvalues and Eigenvectors
Eigenvalues represent the scaling factors of linear transformations, while eigenvectors define the invariant directions. Mastery of the characteristic equation is essential for solving these problems quickly.
- Characteristic Equation: Det(A - λI) = 0
- Sum of Eigenvalues = Trace(A)
- Product of Eigenvalues = Determinant(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(kA) = k^n * Det(A)
A * adj(A) = Det(A) * I
A^-1 = adj(A) / Det(A)
Rank(A) <= min(m, n)
Sum(λ_i) = Trace(A)
Product(λ_i) = Det(A)
A^2 - (trace A)A + (det A)I = 0 (for 2x2 matrix)
Exam Tip
Always use the trace and determinant properties to verify your calculated eigenvalues before proceeding to find eigenvectors.
Common Mistakes
- Confusing the properties of determinant of kA as k*Det(A) instead of k^n*Det(A) where n is the order.
- Neglecting the trivial solution case (X=0) when calculating rank for homogeneous systems.
- Forgetting to verify if a matrix is square before attempting to find eigenvalues.
More Revision Notes
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