MA 35100 Lecture Notes - Lecture 29: Laplace Expansion, If And Only If, Main Diagonal
Document Summary
Prop 3. 9: if w is a vs. that is isomorphic to an n-dimensional v. s. If ex, /m3 is a basis for and t: v wisan isomorphism, then [tcx][(xn)3 is a basis for w. In general, every square matrix has a number associatedto it called its determinant. 3. 71 says a is invertible iff get (a) fc. Goal is to explain why this holds for all square matricies a. If a is 2x2, a= [89], detca)=ad-bc ~common. If a is xh, det (a) is defined using cofactor expansion, given along a vow: r down a in terms of determinants of (n-1) x (n-1) matricies. If a = (aij], the cofactor of (aij] is a scalar. Cij=(-1)"3 deleting row is column; from a. det (aij) n (n-17 x (n-1) matrix obtained from a by cofactor expansion along the ith row of atofind the detca): det (a) = a;10;1 + 9i2972+ @incin.
