3.1 Introduction to DeterminantsNotation: Aijis the matrix obtained from matrix A by deleting theith row and jth column of A.EXAMPLE:A =123456789 10 11 1213 14 15 16A23=Recall that detabcd= ad − bc and we let deta= a.For n ≥ 2,thedeterminant of an n × n matrix A =aijis givenbydet A = a11det A11− a12det A12+ ⋯ +−11+na1ndet A1n=∑j=1n−11+ja1jdet A1j1EXAMPLE: Compute the determinant of A =1203 −12201Solutiondet A = 1 det−1201− 2 det3221+ 0 det3 −120= ______________________________ = ______Common notation: det3221=3221.So1203 −12201= 1−1201− 23221+ 03 −120The i, j-cofactor of A is the number CijwhereCij=−1i+jdet Aij.1203 −12201= 1C11+ 2C12+ 0C13(cofactor expansion across row 1)2THEOREM 1 The determinant of an n × n matrix A can becomputed by a cofactor expansion across any row or down anycolumn:det A = ai1Ci1+ ai2Ci2+ ⋯ + ainCin(expansion across row i)det A = a1jC1j+ a2jC2j+ ⋯ + anjCnj(expansion downcolumn j)Use a matrix of signs to determine−1i+j+ − + ⋯− + − ⋯+ − + ⋯⋮⋮⋮⋱EXAMPLE: Compute the determinant of A =1203 −12201using cofactor expansion down column 3.Solution1203 −12201= 03 −120− 21220+ 1123 −1= 1.3EXAMPLE: Compute the determinant of A =1234021500210035Solution1234021500210035= 1215021035− 0234021035+ 0234215035− 0234215021= 1 ⋅ 22135= 14Method of cofactor expansion is not practical for large matrices -see Numerical Note on page 190.4Triangular Matrices:∗∗⋯ ∗∗0 ∗ ⋯ ∗∗00⋱ ∗∗00 0 ∗∗00 0 0∗∗ 0000∗∗ 000∗∗⋱ 00∗∗⋯ ∗ 0∗∗⋯ ∗∗(upper triangular) (lower triangular)THEOREM 2:IfA is a triangular matrix, then det A is the productof the main diagonal entries of A.EXAMPLE:23 4 501 2 300−3500 0 4= _____________________ = −
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