§7.1Eigenvalues andEigenvectorsEigenvalues andEigenvectorsThe CharacteristicPolynomialFinding EigenvectorsImportantINotice that we had to row reduce a completely differentsingular matrix for each eigenvalue.IAt no time, did we row reduce the original matrixA. Weshould NEVER have to row reduceAitself at all to findeigenvalues or eigenvectors. Row reducingAin an effort to“simplify” the work will throw the answer off. Rowequivalent matrices DO NOT share eigenvalues andeigenvectors.IThe matrix we end up row reducingλIn-Awill ALWAYSend up being singular. That is, the system(λIn-A)x=0will always have (infinitely many) nontrivial solutions.