Linear Algebra Topics - GATE CS & IT Syllabus

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Operations on matrices: addition, multiplication, transpose, inverse, rank, and properties.

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📌 Key Formuladet(AB)=det(A)det(B), (A^{-1})^{-1}=A, (A^T)^{-1}=(A^{-1})^T.

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Scalars λ such that Av = λv for some non-zero vector v.

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📌 Key FormulaCharacteristic equation: det(A - λI)=0, Sum of eigenvalues = trace(A), Product = det(A).

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Non-zero vectors that change only by a scalar factor when a linear transformation is applied.

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📌 Key FormulaFor eigenvalue λ, solve (A-λI)v=0.

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Sets of linear equations represented as Ax = b.

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📌 Key FormulaConsistency: rank(A) = rank(A|b). Unique solution iff rank = number of unknowns.

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Scalar value that encodes linear transformation properties.

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📌 Key Formuladet(AB)=det(A)det(B), det(A^T)=det(A), det(A^{-1})=1/det(A).

6

Factorizing a matrix into lower triangular (L) and upper triangular (U).

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📌 Key FormulaA = L U, where L has ones on diagonal.