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

2016/6/4 Elementary Linear Algebra 2 Chapter Content Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants

2016/6/4 Elementary Linear Algebra Minor and Cofactor Definition  Let A be m  n The (i,j)-minor of A, denoted M ij 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 C ij, is (-1) i+j M ij Remark  Note that C ij =  M ij and the signs (-1) i+j in the definition of cofactor form a checkerboard pattern:

2016/6/4 Elementary Linear Algebra Example 1 Let The minor of entry a 11 is The cofactor of a 11 is C 11 = (-1) 1+1 M 11 = M 11 = 16 Similarly, the minor of entry a 32 is The cofactor of a 32 is C 32 = (-1) 3+2 M 32 = -M 32 = -26

2016/6/4 Elementary Linear Algebra Cofactor Expansion The definition of a 3×3 determinant in terms of minors and cofactors  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

2-1 Cofactor Expansion 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 Theorem (Expansions by Cofactors)  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 1j C 1j + a 2j C 2j +… + a nj C nj (cofactor expansion along the jth column) and det(A) = a i1 C i1 + a i2 C i2 +… + a in C in (cofactor expansion along the ith row) 2016/6/4 Elementary Linear Algebra 6

2-1 Example 3 & 4 Example 3  cofactor expansion along the first column of A Example 4  smart choice of row or column  det(A) = ? 2016/6/4 Elementary Linear Algebra 7

2016/6/4 Elementary Linear Algebra Adjoint of a Matrix If A is any n  n matrix and C ij is the cofactor of a ij, 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) Remarks  If one multiplies the entries in any row by the corresponding cofactors from a different row, the sum of these products is always zero.

2-1 Example 5 Let  a 11 C 31 + a 12 C 32 + a 13 C 33 = ?  Let 2016/6/4 Elementary Linear Algebra 9

2016/6/4 Elementary Linear Algebra Example 6 & 7 Let 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 The matrix of cofactor and adjoint of A are The inverse (see below) is

2016/6/4 Elementary Linear Algebra 11 Theorem (Inverse of a Matrix using its Adjoint) If A is an invertible matrix, then  Show first that

Theorem 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;  det(A) = a 11 a 22 …a nn  E.g. 2016/6/4 Elementary Linear Algebra 12

2-1 Prove Theorem 1.7.1c A triangular matrix is invertible if and only if its diagonal entries are all nonzero 2016/6/4 Elementary Linear Algebra 13

2-1 Prove Theorem 1.7.1d The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular 2016/6/4 Elementary Linear Algebra 14

Theorem (Cramer’s Rule) 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 A j is the matrix obtained by replacing the entries in the column of A by the entries in the matrix b = [b 1 b 2 ··· b n ] T 2016/6/4 Elementary Linear Algebra 15

2016/6/4 Elementary Linear Algebra Example 9 Use Cramer’s rule to solve Since Thus,

2016/6/4 Elementary Linear Algebra 17 Chapter Content Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants

2016/6/4 Elementary Linear Algebra 18 Theorems Theorem  Let A be a square matrix If A has a row of zeros or a column of zeros, then det(A) = 0. Theorem Let A be a square matrix  det(A) = det(A T )

2016/6/4 Elementary Linear Algebra 19 Theorem (Elementary Row Operations) Let A be an n  n matrix  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

2-2 Example of Theorem /6/4 Elementary Linear Algebra 20

2016/6/4 Elementary Linear Algebra 21 Theorem (Elementary Matrices) Let E be an n  n elementary matrix  If E results from multiplying a row of I n by k, then det(E) = k  If E results from interchanging two rows of I n, then det(E) = -1  If E results from adding a multiple of one row of I n to another, then det(E) = 1 Example 2

Theorem (Matrices with Proportional Rows or Columns) If A is a square matrix with two proportional rows or two proportional column, then det(A) = 0 Example /6/4 Elementary Linear Algebra 22

2016/6/4 Elementary Linear Algebra Example 4 (Using Row Reduction to Evaluate a Determinant) Evaluate det(A) where 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

2016/6/4 Elementary Linear Algebra 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.

2-2 Example 5 Using column operation to evaluate a determinant  Compute the determinant of 2016/6/4 Elementary Linear Algebra 25

2-2 Example 6 Row operations and cofactor expansion  Compute the determinant of 2016/6/4 Elementary Linear Algebra 26

2016/6/4 Elementary Linear Algebra 27 Chapter Content Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants

2016/6/4 Elementary Linear Algebra Basic Properties of Determinant 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 kA has a common factor of k, we obtain det(kA) = k n det(A) 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).

2-3 Example 1 det(A+B) ≠det(A)+det(B) 2016/6/4 Elementary Linear Algebra 29

2016/6/4 Elementary Linear Algebra 30 Theorems 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

2-3 Example 2 Using Theorem /6/4 Elementary Linear Algebra 31

2016/6/4 Elementary Linear Algebra 32 Lemma If B is an n  n matrix and E is an n  n elementary matrix, then det(EB) = det(E) det(B) Remark:  If B is an n  n matrix and E 1, E 2, …, E r, are n  n elementary matrices, then det(E 1 E 2 · · · E r B) = det(E 1 ) det(E 2 ) · · · det(E r ) det(B)

Theorem (Determinant Test for Invertibility) A square matrix A is invertible if and only if det(A)  0 Example /6/4 Elementary Linear Algebra 33

2016/6/4 Elementary Linear Algebra 34 Theorem If A and B are square matrices of the same size, then det(AB) = det(A) det(B) Example 4

Theorem If A is invertible, then 2016/6/4 Elementary Linear Algebra 35

2016/6/4 Elementary Linear Algebra Linear Systems of the Form Ax = x 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 Such systems are really homogeneous linear in disguise, since the expresses can be rewritten as ( I – A)x = 0

2016/6/4 Elementary Linear Algebra Eigenvalue and Eigenvector 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.

2-3 Eigenvalue and Eigenvector Remark:  A primary problem of linear system ( I – A)x = 0 is to determine those values of for which the system has a nontrivial solution. Theorem (Eigenvalues and Singularity)  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) 2016/6/4 Elementary Linear Algebra 38

2016/6/4 Elementary Linear Algebra Example 5 & 6 The linear system The characteristic equation of A is 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 = [3t/4 t] T – the other eigenvector

2016/6/4 Elementary Linear Algebra 40 Theorem (Equivalent Statements) If A is an n  n matrix, then the following are equivalent  A is invertible.  Ax = 0 has only the trivial solution  The reduced row-echelon form of A as I n  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

2016/6/4 Elementary Linear Algebra Permutation A permutation of the set of integers {1,2,…,n} is an arrangement of these integers in some order without omission repetition Example 1  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  List all permutations of the set of integers {1,2,3,4}

2016/6/4 Elementary Linear Algebra Inversion An inversion is said to occur in a permutation (j 1, j 2, …, j n ) whenever a larger integer precedes a smaller one. The total number of inversions occurring in a permutation can be obtained as follows: 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, …, j n. The sum of these number will be the total number of inversions in the permutation

2016/6/4 Elementary Linear Algebra Example 3 Determine the number of inversions in the following permutations:  (6,1,3,4,5,2)  (2,4,1,3)  (1,2,3,4) Solution:  The number of inversions is = 8  The number of inversions is = 3  There no inversions in this permutation

2016/6/4 Elementary Linear Algebra Classifying Permutations 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 Example 4  The following table classifies the various permutations of {1,2,3} as even or odd PermutationNumber of Inversions classification (1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1) even odd even odd

2016/6/4 Elementary Linear Algebra Elementary Product 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  The elementary product of the matrix is

2016/6/4 Elementary Linear Algebra Signed Elementary Product An n  n matrix A has n! elementary products. There are the products of the form a 1j 1 a 2j 2 ··· a nj n, where (j 1, j 2, …, j n ) is a permutation of the set {1, 2, …, n}. By a signed elementary product from A we shall mean an elementary a 1j 1 a 2j 2 ··· a nj n multiplied by +1 or -1.  We use + if (j 1, j 2, …, j n ) is an even permutation and – if (j 1, j 2, …, j n ) is an odd permutation

2016/6/4 Elementary Linear Algebra Example 6 List all signed elementary products from the matrices Elementary Product Associated Permutation Even or OddSigned 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 OddSigned 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 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

2016/6/4 Elementary Linear Algebra Determinant 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

2016/6/4 Elementary Linear Algebra Using mnemonic for Determinant The determinant is computed by summing the products on the rightward arrows and subtracting the products on the leftward arrows Remark:  This method will not work for determinant of 4  4 matrices or higher!

2-4 Example 8 Evaluate the determinants of 2016/6/4 Elementary Linear Algebra 50

2016/6/4 Elementary Linear Algebra Notation and Terminology 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 1j 1 a 2j 2 ··· a nj n where  indicates that the terms are to be summed over all permutations (j 1, j 2, …, j n ) and the + or – is selected in each term according to where the permutation is even or odd Remark:  4  4 matrices need 4! = 24 signed elementary products  10  10 determinant need 10! = signed elementary products!  Other methods are required.