Final Cheat Sheet PDF

Row Reduction Determinant & Cramer's , Rule Eigenvalues/vectors...

Row Reduction Determinant & Cramer's , Rule Eigenvalues/vectors Diagonalization & only if non-defective Adjoint un] 1 (iii) S 2v v 1) find matrix of cotactors I = A I y.... , 102 2y , = = , S"AS diay 21 Xc 1n] yz 2) Transpose Mc =., 4400 ,.... y 2yc by3 = + + , 5 300 503 y's = Yz - 3y3 A = eigenvalues of A corresponding to v * A " = eCA) adj (A) * (ij 1)i j(M)ij + using C, ( = - SAS= D where D diag(X Da xn) det (A) = AnCn + AziCei + AziCs D Write the DE's as a matrix = , ,... -( 1) - -)=%/ = = (* !] 1) Determine He eigenvalues of the matrix [ = ]k 2) construct diagonal D diagonal det(A) = 4024-(-4020) = 4 2) Find Matrix using A., 1. In along main eigenvalues - - , , det(Bx) 3) Determine linearly independent eigenvectors I I Cramers Rule : 21 : det(A-x1) det C * a 4) - det (A) = I 1 I 100 101 A replace var , with RHS Combine He column rectors from step 3 to gets ↳ 40 400 0 -s - x 5) Find S "doing [S1 ! :] using Ri 500 5 300 503 def(A-x1) = AnCn+AlzCiz + Ais[13 6) Put in form : A=SDS" or S"AS = $ x2) =x) * - I det (B ) 101 102 100 (2 x) Defective vs Nondetective EeC = = - - , , 400408482 =(2 - x)[(2- x)3-x) 3] - nondelective - non matrix that has a linearly independent eigenvectors =(2 - x)) - b + x + x - b) detective - A has less than n linearly independent eigenvectors I defective vectors are basis (2 x))x2 eigenbasis it A is those a = non - - + x - 12) , =(2 - x)(x + 4)(X - 3) nondelective means dim (Ei] = multiplicity is , if dim (Ei]