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Understanding Determinants in Linear Algebra: Properties and Applications

Explore the properties of determinants for square matrices, including exchanging rows, equal rows, zero rows, and the relationship with LU factorization. Learn how determinants signal singular or invertible matrices.

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Understanding Determinants in Linear Algebra: Properties and Applications

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  1. Lecture 18 • Determinants det(A) square matrix • Properties Linear Algebra---Meiling CHEN

  2. Properties : • Exchange rows reverse sign of det • 2 equal rows  det=0 (exchange rows get same matrix only probability is 0) Linear for each row Linear Algebra---Meiling CHEN

  3. Subtract l*row i from row j det does not change • 6 row of zeros  det(A)=0 • 7. Product of Pivots One of advantage of LU factorization By elimination we can kill these Linear Algebra---Meiling CHEN

  4. Zero row • 8. when A is singular • when A is invertible • 9 • 10. A=LU Linear Algebra---Meiling CHEN

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