3.1 Introduction Math 2331 { Linear Algebra 3.1 Introduction to Determinants Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices 3.1 Introduction to Determinants Determinant via Cofactor Expansion Examples Theorem Triangular Matrices Examples Theorem Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Introduction to Determinants Notation: Aij is the matrix obtained from matrix A by deleting the ith row and jth column of A. Example 2 1 2 3 4 3 2 3 6 5 6 7 8 7 6 7 A = 6 7 A23 = 6 7 4 9 10 11 12 5 4 5 13 14 15 16 a b Recall that det = ad − bc and we let det [a] = a. c d For n ≥ 2, the determinant of an n × n matrix A = [aij ] is 1+n det A = a11 det A11 − a12 det A12 + ··· + (−1) a1n det A1n n P 1+j = (−1) a1j det A1j j=1 Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Determinants: Example Example 2 1 2 0 3 Compute the determinant of A = 4 3 −1 2 5 2 0 1 Solution −1 2 3 2 3 −1 det A = 1 det − 2 det + 0 det 0 1 2 1 2 0 = = 3 2 3 2 Common notation: det = : 2 1 2 1 So 1 2 0 −1 2 3 2 3 −1 3 −1 2 = 1 − 2 + 0 0 1 2 1 2 0 2 0 1 Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Determinants: Cofactor Expansion Cofactor The (i; j)-cofactor of A is the number Cij where i+j Cij = (−1) det Aij : Example (Cofactor Expansion) 1 2 0 3 −1 2 = 1C11 + 2C12 + 0C13 2 0 1 (cofactor expansion across row 1) Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Cofactor Expansion: Theorem Theorem (Cofactor Expansion) The determinant of an n × n matrix A can be computed by a cofactor expansion across any row or down any column: det A = ai1Ci1 + ai2Ci2 + ··· + ainCin (expansion across row i) det A = a1j C1j + a2j C2j + ··· + anj Cnj (expansion down column j) Use a matrix of signs to determine (−1)i+j 2 + − + ··· 3 6 − + − · · · 7 6 7 6 + − + ··· 7 4 . 5 . .. Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Cofactor Expansion: Example Example 2 1 2 0 3 Compute the determinant of A = 4 3 −1 2 5 using cofactor 2 0 1 expansion down column 3. Solution 1 2 0 3 −1 1 2 1 2 3 −1 2 = 0 − 2 + 1 = 1. 2 0 2 0 3 −1 2 0 1 Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Cofactor Expansion: Example Example 2 1 2 3 4 3 6 0 2 1 5 7 Compute the determinant of A = 6 7 4 0 0 2 1 5 0 0 3 5 Solution: 1 2 3 4 0 2 1 5 0 0 2 1 0 0 3 5 2 1 5 2 3 4 2 3 4 2 3 4 = 1 0 2 1 − 0 0 2 1 + 0 2 1 5 − 0 2 1 5 0 3 5 0 3 5 0 3 5 0 2 1 2 1 = 1 · 2 = 14 3 5 Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Triangular Matrices Method of cofactor expansion is not practical for large matrices - see Numerical Note on page 167. Triangular Matrices 2 ∗ ∗ · · · ∗ ∗ 3 2 ∗ 0 0 0 0 3 6 0 ∗ · · · ∗ ∗ 7 6 ∗ ∗ 0 0 0 7 6 7 6 7 6 .. 7 6 .. 7 6 0 0 . ∗ ∗ 7 6 ∗ ∗ . 0 0 7 6 7 6 7 4 0 0 0 ∗ ∗ 5 4 ∗ ∗ · · · ∗ 0 5 0 0 0 0 ∗ ∗ ∗ · · · ∗ ∗ (upper triangular) (lower triangular) Theorem If A is a triangular matrix, then det A is the product of the main diagonal entries of A. Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 10 3.1 Introduction Definition Cofactor Expansion Triangular Matrices Triangular Matrices: Example Example 2 3 4 5 0 1 2 3 = = −24 0 0 −3 5 0 0 0 4 Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 10.