12 1. Preliminaries (2) Exchanging two rows, or two columns, of A changes the sign of the determi- nant. (3) If we multiply a row or column of A by a nonzero constant c, then the deter- minant of the resulting matrix is c det(A). (4) The determinant of a triangular matrix is the product of the diagonal entries. Another important determinant formula is the formula for expansion by cofac- tors. For an n × n matrix A, let Aij denote the (n − 1) × (n − 1) matrix which remains after removing row i and column j from A. Then, for any i = 1,...,n, we have (1.8) det(A) = n k=1 (−1)i+kaik det(Aik). The quantities (−1)i+k det(Aik) are called cofactors, and formula (1.8) is the La- grange expansion by cofactors. Formula (1.8) gives the expansion along row i. A similar result holds for columns the formula for cofactor expansion along column j is (1.9) det(A) = n k=1 (−1)k+jakj det(Akj). In these formulas, the cofactor (−1)i+k det(Aik) is multiplied by the entry aik, of the same position. If we use the cofactors of entries in row i but multiply them by entries from a different row, then the sum is zero. Thus, if i = r, then (1.10) n k=1 (−1)r+kaik det(Ark) = 0. A similar formula holds for columns: if j = r, then (1.11) n k=1 (−1)k+rakj det(Akr) = 0. Letting Cof(A) denote the n × n matrix with (−1)i+j det(Aij) in position i, j, formulas (1.8) and (1.10) tell us A(Cof(A))T = (det A)In. When det(A) = 0, we have (1.12) A−1 = 1 det A (Cof(A))T . Formula (1.12) is rarely a good way to compute A−1, but it can give useful qualita- tive information about the entries of A−1. For example, if A is a matrix of integers and det A = ±1, then formula (1.12) shows that A−1 has integer entries. Finally, a few other important facts should be familiar to the reader: (1) An n × n matrix A is invertible if and only if det(A) = 0. (2) If A and B are n × n matrices, then det(AB) = det(A) det(B). (3) If A is an n × n matrix, then det A = det(AT ).

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