Questions
2 questions per PSU paper
Difficulty
Medium
Importance
High yield for HPCL/NTPC/ONGC
Overview
Linear Algebra is a foundational pillar of Engineering Mathematics, focusing on the manipulation of matrices and the solution of linear systems. It is highly significant for PSU exams as it provides a predictable source of marks through algorithmic problems involving eigenvalues and matrix properties. Aspirants must master the mechanics of row operations and the characteristic equation to succeed.
Matrix Algebra & Rank
Matrix operations form the basic building blocks, while the rank of a matrix determines the dimensionality of its row or column space. Understanding rank is essential for predicting the solvability of system equations.
- Rank(A) is the maximum number of linearly independent rows or columns.
- Rank(A) = Rank(transpose of A).
- Rank(AB) <= min(Rank(A), Rank(B)).
- A square matrix is singular if its determinant is zero, meaning Rank < n.
Systems of Linear Equations
This subtopic deals with finding solutions to AX = B. The consistency of the system depends on the comparison of the rank of the coefficient matrix A and the augmented matrix [A|B].
- Consistent system if Rank(A) = Rank(A|B).
- Unique solution if Rank(A) = Rank(A|B) = n (number of unknowns).
- Infinite solutions if Rank(A) = Rank(A|B) < n.
- Inconsistent system (no solution) if Rank(A) != Rank(A|B).
Eigenvalues & Eigenvectors
Eigenvalues are scalars that characterize the transformation of a linear system, found using the characteristic equation. They are frequently tested in PSU exams due to their direct algebraic application.
- Characteristic equation: det(A - lambda*I) = 0.
- Sum of eigenvalues = Trace of matrix (sum of diagonal elements).
- Product of eigenvalues = Determinant of the matrix.
- Eigenvalues of a triangular matrix are its diagonal elements.
Cayley–Hamilton Theorem
This theorem states that every square matrix satisfies its own characteristic equation. It is a powerful time-saving tool for finding the inverse of a matrix or calculating high powers of a matrix without direct multiplication.
- Every matrix satisfies: P(lambda) = det(A - lambda*I) = 0.
- Use to find A inverse by multiplying the equation by A^-1.
- Highly useful for computing A^n for large n.
- Reduces complex matrix polynomials to simpler forms.
Formula Sheet
det(A - lambda*I) = 0
Sum(lambda_i) = Trace(A)
Product(lambda_i) = det(A)
Rank(A) = Rank(A^T)
AX = B (Consistent if Rank(A) = Rank(A|B))
A^-1 = [1/det(A)] * adj(A)
A^n using Cayley-Hamilton: Replace lambda with A in characteristic equation
Exam Tip
When asked for A inverse using Cayley-Hamilton, always rearrange the characteristic equation to isolate the identity matrix term to avoid tedious manual inversion.
Common Mistakes
- Confusing the condition for unique vs. infinite solutions in linear systems.
- Forgetting to include the negative sign when calculating the characteristic polynomial.
- Assuming that Rank(AB) = Rank(A) * Rank(B), which is incorrect.
More Revision Notes
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