MATH 355 Lecture 10: Determinants and Jacobians Math 410
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Once more: determinants: definition: suppose we have a square matrix a", volume is a signed quantity here: Example: suppose f(x) is a smooth function with property f(lambda*x)=lambda*f(x) then f(x)=a*x: proof: Important observation that will allow us to give determinant for arbitrary number of dimensions. The most efficient way to compute determinants: plu factorization to solve for determinants. How do we compute determinants: 2d: check, 3d: check, general case: find plu factorization, pa=lu, use signature of p and u det(a)=(signature of p)*(product of (main diagonal only) diagonal entries of u) Laplace"s rule for expansion with respect to rows/columns: j can be anything. Example 2: note: we may expand over a row instead of a column, basically we can switch i and j in the formula (see laplace"s rule above) Why laplace"s rule (see first slide below) works: proof: not important to know, not being tested on, signature will not change for a11 term.