Math 2331 – Linear Algebra 3.1 Introduction to Determinants

3.1 Introduction

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  1 2 3 4     5 6 7 8    A =   A23 =    9 10 11 12    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  1 2 0  Compute the determinant of A =  3 −1 2  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

 + − + ···   − + − · · ·     + − + ···   . . . .  . . . ..

Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 10 3.1 Introduction Deﬁnition Cofactor Expansion Triangular Matrices Cofactor Expansion: Example

Example  1 2 0  Compute the determinant of A =  3 −1 2  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  1 2 3 4   0 2 1 5  Compute the determinant of A =    0 0 2 1  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  ∗ ∗ · · · ∗ ∗   ∗ 0 0 0 0   0 ∗ · · · ∗ ∗   ∗ ∗ 0 0 0       ..   ..   0 0 . ∗ ∗   ∗ ∗ . 0 0       0 0 0 ∗ ∗   ∗ ∗ · · · ∗ 0  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