16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-232. The Fundamental Theorem of Algebra is an existence theorem. It givesno information about how to find the roots of a given polynomial z(r).For degree two, one can explicitly find the roots via that quadraticformula (i.e., involving square roots of expressions involving the coefficients).For degrees three and four, there also are explicit formulas to find theroots in terms of taking expressions involving various roots of the coefficients.For degree greater than four, there are no such formulas thatwork in all cases. This surprising result (often called the unsolvabilityof the quintic) was proved by Abel in 1823. For more information lookup the Abel-Ruffini Theorem on Wikipedia.Note that some texts call the polynomial z 1 (r) = det(A − rI) the characteristicpolynomial of A. Since z 1 (r) = (−1) n z(r), these two polynomialshave the same roots, so to solve problems concerning eigenvalues, it reallydoes not matter which definition is used.Of course, even if A is real, the characteristic polynomial z(r) may havereal or complex roots. A real eigenvalue will have associated eigenvectorswhich are also real, and a complex eigenvalue will only have associated complexeigenvectors (i.e. written as u 1 + iu 2 with u 1 and u 2 both real vectorsand u 2 non-zero).6.1 Simple formulas for the characteristic polynomialsof 2 × 2 and 3 × 3 matricesFor any matrix A = (a ij ), define the trace of A to ben∑tr(A) = a iii=1Thus, tr(A) is simply the sum of the diagonal entries.If( )a bA = ,c dthen