Elementary Linear Algebra Anton Rorres 9 th Edition

Elementary Linear Algebra Anton & Rorres, 9 th Edition Lecture Set – 02 Chapter 2: Determinants

Chapter Content n n Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants 2021/2/24 Elementary Linear Algebra 2

2 -1 Minor and Cofactor n Definition q Let A be m n n The (i, j)-minor of A, denoted Mij is the determinant of the (n-1) (n 1) matrix formed by deleting the ith row and jth column from A The (i, j)-cofactor of A, denoted Cij, is (-1)i+j Mij Remark q 2021/2/24 Note that Cij = Mij and the signs (-1)i+j in the definition of cofactor form a checkerboard pattern: Elementary Linear Algebra 3

2 -1 Example 1 n Let n The minor of entry a 11 is n The cofactor of a 11 is C 11 = (-1)1+1 M 11 = 16 n Similarly, the minor of entry a 32 is n The cofactor of a 32 is C 32 = (-1)3+2 M 32 = -26 2021/2/24 Elementary Linear Algebra 4

2 -1 Cofactor Expansion n The definition of a 3× 3 determinant in terms of minors and cofactors q det(A) = a 11 M 11 +a 12(-M 12)+a 13 M 13 = a 11 C 11 +a 12 C 12+a 13 C 13 this method is called cofactor expansion along the first row of A Example 2 q n 2021/2/24 Elementary Linear Algebra 5

2 -1 Cofactor Expansion n det(A) =a 11 C 11 +a 12 C 12+a 13 C 13 = a 11 C 11 +a 21 C 21+a 31 C 31 =a 21 C 21 +a 22 C 22+a 23 C 23 = a 12 C 12 +a 22 C 22+a 32 C 32 =a 31 C 31 +a 32 C 32+a 33 C 33 = a 13 C 13 +a 23 C 23+a 33 C 33 n Theorem 2. 1. 1 (Expansions by Cofactors) q 2021/2/24 The determinant of an n n matrix A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 i, j n det(A) = a 1 j. C 1 j + a 2 j. C 2 j +… + anj. Cnj (cofactor expansion along the jth column) and det(A) = ai 1 Ci 1 + ai 2 Ci 2 +… + ain. Cin (cofactor expansion along the ith row) Elementary Linear Algebra 6

2 -1 Example 3 & 4 n Example 3 q n cofactor expansion along the first column of A Example 4 q smart choice of row or column q det(A) = ? 2021/2/24 Elementary Linear Algebra 7

2 -1 Adjoint of a Matrix n If A is any n n matrix and Cij is the cofactor of aij, then the matrix is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted by adj(A) n Remarks q 2021/2/24 If one multiplies the entries in any row by the corresponding cofactors from a different row, the sum of these products is always zero. Elementary Linear Algebra 8

2 -1 Example 5 n Let a 11 C 31 + a 12 C 32 + a 13 C 33 = ? q q Let 2021/2/24 Elementary Linear Algebra 9

2 -1 Example 6 & 7 n Let n The cofactors of A are: C 11 = 12, C 12 = 6, C 13 = -16, C 21 = 4, C 22 = 2, C 23 = 16, C 31 = 12, C 32 = -10, C 33 = 16 n The matrix of cofactor and adjoint of A are n The inverse (see below) is 2021/2/24 Elementary Linear Algebra 10

Theorem 2. 1. 2 (Inverse of a Matrix using its n If A is an invertible matrix, then Adjoint) q 2021/2/24 Show first that Elementary Linear Algebra 11

Theorem 2. 1. 3 n If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix; q q 2021/2/24 det(A) = a 11 a 22…ann E. g. Elementary Linear Algebra 12

2 -1 Prove Theorem 1. 7. 1 c n A triangular matrix is invertible if and only if its diagonal entries are all nonzero 2021/2/24 Elementary Linear Algebra 13

2 -1 Prove Theorem 1. 7. 1 d n The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular 2021/2/24 Elementary Linear Algebra 14

Theorem 2. 1. 4 (Cramer’s Rule) n If Ax = b is a system of n linear equations in n unknowns such that det( I – A) 0 , then the system has a unique solution. This solution is where Aj is the matrix obtained by replacing the entries in the column of A by the entries in the matrix b = [b 1 b 2 ··· bn]T 2021/2/24 Elementary Linear Algebra 15

2 -1 Example 9 n Use Cramer’s rule to solve n Since n Thus, 2021/2/24 Elementary Linear Algebra 16

Chapter Content n n Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants 2021/2/24 Elementary Linear Algebra 17

Theorems n Theorem 2. 2. 1 q Let A be a square matrix n If A has a row of zeros or a column of zeros, then det(A) = 0. n 2021/2/24 Theorem 2. 2. 2 n Let A be a square matrix q det(A) = det(AT) Elementary Linear Algebra 18

Theorem 2. 2. 3 (Elementary Row Operations) n Let A be an n n matrix q q q n If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, than det(B) = k det(A) If B is the matrix that results when two rows or two columns of A are interchanged, then det(B) = - det(A) If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple column is added to another column, then det(B) = det(A) Example 1 2021/2/24 Elementary Linear Algebra 19

2 -2 Example of Theorem 2. 2. 3 2021/2/24 Elementary Linear Algebra 20

Theorem 2. 2. 4 (Elementary Matrices) n Let E be an n n elementary matrix q q q n If E results from multiplying a row of In by k, then det(E) = k If E results from interchanging two rows of In, then det(E) = -1 If E results from adding a multiple of one row of In to another, then det(E) = 1 Example 2 2021/2/24 Elementary Linear Algebra 21

Theorem 2. 2. 5 (Matrices with Proportional Rows or Columns) n If A is a square matrix with two proportional rows or two proportional column, then det(A) = 0 n Example 3 2021/2/24 Elementary Linear Algebra 22

2 -2 Example 4 (Using Row Reduction to Evaluate a Determinant) n Evaluate det(A) where n Solution: The first and second rows of A are interchanged. A common factor of 3 from the first row was taken through the determinant sign 2021/2/24 Elementary Linear Algebra 23

2 -2 Example 4 (continue) -2 times the first row was added to the third row. -10 times the second row was added to the third row A common factor of -55 from the last row was taken through the determinant sign. 2021/2/24 Elementary Linear Algebra 24

2 -2 Example 5 n Using column operation to evaluate a determinant q 2021/2/24 Compute the determinant of Elementary Linear Algebra 25

2 -2 Example 6 n Row operations and cofactor expansion q 2021/2/24 Compute the determinant of Elementary Linear Algebra 26

Chapter Content n n Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants 2021/2/24 Elementary Linear Algebra 27

2 -3 Basic Properties of Determinant n Since a common factor of any row of a matrix can be moved through the det sign, and since each of the n row in k. A has a common factor of k, we obtain det(k. A) = kndet(A) n There is no simple relationship exists between det(A), det(B), and det(A+B) in general. In particular, we emphasize that det(A+B) is usually not equal to det(A) + det(B). n 2021/2/24 Elementary Linear Algebra 28

2 -3 Example 1 n det(A+B) ≠det(A)+det(B) 2021/2/24 Elementary Linear Algebra 29

Theorems 2. 3. 1 n n Let A, B, and C be n n matrices that differ only in a single row, say the r-th, and assume that the r-th row of C can be obtained by adding corresponding entries in the r-th rows of A and B. Then det(C) = det(A) + det(B) The same result holds for columns. Let 2021/2/24 Elementary Linear Algebra 30

2 -3 Example 2 n Using Theorem 2. 3. 1 2021/2/24 Elementary Linear Algebra 31

Lemma 2. 3. 2 n If B is an n n matrix and E is an n n elementary matrix, then det(EB) = det(E) det(B) n Remark: q 2021/2/24 If B is an n n matrix and E 1, E 2, …, Er, are n n elementary matrices, then det(E 1 E 2 · · · Er B) = det(E 1) det(E 2) · · · det(Er) det(B) Elementary Linear Algebra 32

Theorem 2. 3. 3 (Determinant Test for Invertibility) n A square matrix A is invertible if and only if det(A) 0 n Example 3 2021/2/24 Elementary Linear Algebra 33

Theorem 2. 3. 4 n If A and B are square matrices of the same size, then det(AB) = det(A) det(B) n Example 4 2021/2/24 Elementary Linear Algebra 34

Theorem 2. 3. 5 n If A is invertible, then 2021/2/24 Elementary Linear Algebra 35

2 -4 Linear Systems of the Form Ax = x n Many applications of linear algebra are concerned with systems of n linear equations in n unknowns that are expressed in the form Ax = x, where is a scalar n Such systems are really homogeneous linear in disguise, since the expresses can be rewritten as ( I – A)x = 0 2021/2/24 Elementary Linear Algebra 36

2 -3 Eigenvalue and Eigenvector n n n The eigenvalues of an n n matrix A are the number for which there is a nonzero x 0 with Ax = x. The eigenvectors of A are the nonzero vectors x 0 for which there is a number with Ax = x. If Ax = x for x 0, then x is an eigenvector associated with the eigenvalue , and vice versa. 2021/2/24 Elementary Linear Algebra 37

2 -3 Eigenvalue and Eigenvector n Remark: q A primary problem of linear system ( I – A)x = 0 is to determine those values of for which the system has a nontrivial solution. n Theorem (Eigenvalues and Singularity) q is an eigenvalue of A if and only if I – A is singular, which in turn holds if and only if the determinant of I – A equals zero: det( I – A) = 0 (the so-called characteristic equation of A) 2021/2/24 Elementary Linear Algebra 38

2 -3 Example 5 & 6 n The linear system n The characteristic equation of A is n n The eigenvalues of A are = -2 and = 5 By definition, x is an eigenvector of A if and only if x is a nontrivial solution of ( I – A)x = 0, i. e. , If = -2, x = [-t t]T – one eigenvector If = 5, x = [3 t/4 t]T – the other eigenvector 2021/2/24 Elementary Linear Algebra 39

Theorem 2. 3. 6 (Equivalent Statements) n If A is an n n matrix, then the following are equivalent q q q q 2021/2/24 A is invertible. Ax = 0 has only the trivial solution The reduced row-echelon form of A as In A is expressible as a product of elementary matrices Ax = b is consistent for every n 1 matrix b Ax = b has exactly one solution for every n 1 matrix b det(A) 0 Elementary Linear Algebra 40

2 -4 Permutation n n A permutation of the set of integers {1, 2, …, n} is an arrangement of these integers in some order without omission repetition Example 1 q n There are six different permutations of the set of integers {1, 2, 3}: (1, 2, 3), (2, 1, 3), (3, 1, 2), (1, 3, 2), (2, 3, 1), (3, 2, 1). Example 2 q List all permutations of the set of integers {1, 2, 3, 4} 2 1 3 2 1 4 3 3 4 1 4 2 1 4 3 2 3 4 2 3 3 4 1 3 2 4 1 2 2 3 1 2 4 3 4 2 3 2 4 3 4 1 3 1 4 2 4 1 2 1 3 2 3 1 2021/2/24 Elementary Linear Algebra 41

2 -4 Inversion n An inversion is said to occur in a permutation (j 1, j 2, …, jn) whenever a larger integer precedes a smaller one. n The total number of inversions occurring in a permutation can be obtained as follows: n n n 2021/2/24 Find the number of integers that are less than j 1 and that follow j 1 in the permutation; Find the number of integers that are less than j 2 and that follow j 2 in the permutation; Continue the process for j 1, j 2, …, jn. The sum of these number will be the total number of inversions in the permutation Elementary Linear Algebra 42

2 -4 Example 3 n Determine the number of inversions in the following permutations: q q q n (6, 1, 3, 4, 5, 2) (2, 4, 1, 3) (1, 2, 3, 4) Solution: q q q 2021/2/24 The number of inversions is 5 + 0 + 1 + 1 = 8 The number of inversions is 1 + 2 + 0 = 3 There no inversions in this permutation Elementary Linear Algebra 43

2 -4 Classifying Permutations n A permutation is called even if the total number of inversions is an even integer and is called odd if the total inversions is an odd integer n Example 4 q The following table classifies the various permutations of {1, 2, 3} as even or odd Permutation Number of classification Inversions (1, 2, 3) (1, 3, 2) (2, 1, 3) (2, 3, 1) (3, 1, 2) (3, 2, 1) 2021/2/24 Elementary Linear Algebra 0 1 1 2 2 3 even odd 44

2 -4 Elementary Product n n By an elementary product from an n n matrix A we shall mean any product of n entries from A, no two of which come from the same row or same column. Example q The elementary product of the matrix is 2021/2/24 Elementary Linear Algebra 45

2 -4 Signed Elementary Product n An n n matrix A has n! elementary products. There are the products of the form a 1 j 1 a 2 j 2 ··· anjn, where (j 1, j 2, …, jn) is a permutation of the set {1, 2, …, n}. n By a signed elementary product from A we shall mean an elementary a 1 j 1 a 2 j 2 ··· anjn multiplied by +1 or -1. q 2021/2/24 We use + if (j 1, j 2, …, jn) is an even permutation and – if (j 1, j 2, …, jn) is an odd permutation Elementary Linear Algebra 46

2 -4 Example 6 n List all signed elementary products from the matrices 2021/2/24 Elementary Product Associated Permutation Even or Odd Signed Elementary Product a 11 a 22 a 12 a 21 (1, 2) (2, 1) even odd a 11 a 22 - a 12 a 21 Elementary Product Associated Permutation Even or Odd Signed Elementary Product a 11 a 22 a 33 a 11 a 23 a 32 a 12 a 21 a 33 a 12 a 23 a 31 a 13 a 21 a 32 a 13 a 22 a 31 (1, 2, 3) (1, 3, 2) (2, 1, 3) (2, 3, 1) (3, 1, 2) (3, 2, 1) even odd a 11 a 22 a 33 - a 11 a 23 a 32 - a 12 a 21 a 33 a 12 a 23 a 31 a 13 a 21 a 32 - a 13 a 22 a 31 Elementary Linear Algebra 47

2 -4 Determinant n n Let A be a square matrix. The determinant function is denoted by det, and we define det(A) to be the sum of all signed elementary products from A. The number det(A) is called the determinant of A Example 7 2021/2/24 Elementary Linear Algebra 48

2 -4 Using mnemonic for Determinant n The determinant is computed by summing the products on the rightward arrows and subtracting the products on the leftward arrows n Remark: q 2021/2/24 This method will not work for determinant of 4 4 matrices or higher! Elementary Linear Algebra 49

2 -4 Example 8 n Evaluate the determinants of 2021/2/24 Elementary Linear Algebra 50

2 -4 Notation and Terminology n n n We note that the symbol |A| is an alternative notation for det(A) The determinant of A is often written symbolically as det(A) = a 1 j 1 a 2 j 2 ··· anjn where indicates that the terms are to be summed over all permutations (j 1, j 2, …, jn) and the + or – is selected in each term according to where the permutation is even or odd Remark: q 4 4 matrices need 4! = 24 signed elementary products q 10 10 determinant need 10! = 3628800 signed elementary products! q Other methods are required. 2021/2/24 Elementary Linear Algebra 51