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EXAMhelp · GATE notes
Linear Algebra
CH 01Matrices and eigenvalues
Key points
- Sum of eigenvalues = trace, product = determinant.
- Rank + nullity = number of columns.
- Ax = b is consistent when rank(A) = rank([A | b]); unique solution if that rank = number of unknowns.
- Eigenvalues of a triangular matrix are its diagonal entries.
- If λ is an eigenvalue of A, then λᵏ is one of Aᵏ and 1/λ is one of A⁻¹.
- Real symmetric matrices have real eigenvalues.
Formulas & facts
- det(AB)
- det(A) · det(B)
- det(kA), A is n × n
- kⁿ · det(A)
- (AB)⁻¹
- B⁻¹A⁻¹