Elementary Linear Algebra Anton & Rorres, 9 th Edition

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

Chapter Content • Determinants by Cofactor Expansion • Evaluating Determinants by Row Reduction • Properties of the Determinant Function • A Combinatorial Approach to Determinants Elementary Linear Algebra

2-1 Minor and Cofactor • Definition • Let A be m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 • Note that Cij = Mij and the signs (-1)i+j in the definition of cofactor form a checkerboard pattern: Elementary Linear Algebra

2-1 Example 1 • Let • The minor of entry a11 is • The cofactor of a11 is C11 = (-1)1+1M11 = M11 = 16 • Similarly, the minor of entry a32 is • The cofactor of a32 is C32 = (-1)3+2M32 = -M32 = -26 Elementary Linear Algebra

2-1 Cofactor Expansion • The definition of a 3×3 determinant in terms of minors and cofactors • det(A) = a11M11 +a12(-M12)+a13M13 = a11C11 +a12C12+a13C13 • this method is called cofactor expansion along the first row of A • Example 2 Elementary Linear Algebra

2-1 Cofactor Expansion • det(A) =a11C11 +a12C12+a13C13 = a11C11 +a21C21+a31C31 =a21C21 +a22C22+a23C23 = a12C12 +a22C22+a32C32 =a31C31 +a32C32+a33C33 = a13C13 +a23C23+a33C33 • Theorem 2.1.1 (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) = a1jC1j + a2jC2j +… + anjCnj (cofactor expansion along the jth column) and det(A) = ai1Ci1 + ai2Ci2+… + ainCin (cofactor expansion along the ith row) Elementary Linear Algebra

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) = ? Elementary Linear Algebra

2-1 Adjoint of a Matrix • If A is any nn matrix and Cij is the cofactor of aij, then the matrixis 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. Elementary Linear Algebra

2-1 Example 5 • Let • a11C31 + a12C32 + a13C33 = ? • Let Elementary Linear Algebra

2-1 Example 6 & 7 • Let • The cofactors of A are: C11 = 12, C12 = 6, C13 = -16, C21 = 4, C22 = 2, C23 = 16, C31 = 12, C32 = -10, C33 = 16 • The matrix of cofactor and adjoint of A are • The inverse (see below) is Elementary Linear Algebra

Theorem 2.1.2 (Inverse of a Matrix using its Adjoint) • If A is an invertible matrix, then • Show first that Elementary Linear Algebra

Theorem 2.1.3 • 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) = a11a22…ann • E.g. Elementary Linear Algebra

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

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 Elementary Linear Algebra

Theorem 2.1.4 (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 Aj is the matrix obtained by replacing the entries in the column of A by the entries in the matrix b = [b1b2··· bn]T Elementary Linear Algebra

2-1 Example 9 • Use Cramer’s rule to solve • Since • Thus, Elementary Linear Algebra

Theorems • Theorem 2.2.1 • Let A be a square matrix • If A has a row of zeros or a column of zeros, then det(A) = 0. • Theorem 2.2.2 • Let A be a square matrix • det(A) = det(AT) Elementary Linear Algebra

Theorem 2.2.3 (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 Elementary Linear Algebra

2-2 Example of Theorem 2.2.3 Elementary Linear Algebra

Theorem 2.2.4 (Elementary Matrices) • Let E be an nn elementary matrix • 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 Elementary Linear Algebra

Theorem 2.2.5 (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 3 Elementary Linear Algebra

2-2 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 Elementary Linear Algebra

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. Elementary Linear Algebra

2-2 Example 5 • Using column operation to evaluate a determinant • Compute the determinant of Elementary Linear Algebra

2-2 Example 6 • Row operations and cofactor expansion • Compute the determinant of Elementary Linear Algebra

2-3 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) = kndet(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). Elementary Linear Algebra

2-3 Example 1 • det(A+B) ≠det(A)+det(B) Elementary Linear Algebra

Theorems 2.3.1 • 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 Elementary Linear Algebra

2-3 Example 2 • Using Theorem 2.3.1 Elementary Linear Algebra

Lemma 2.3.2 • 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 E1, E2, …, Er, are nn elementary matrices, then det(E1 E2 ···Er B) = det(E1) det(E2) ···det(Er) det(B) Elementary Linear Algebra

Theorem 2.3.3 (Determinant Test for Invertibility) • A square matrix A is invertible if and only if det(A)  0 • Example 3 Elementary Linear Algebra

Theorem 2.3.4 • If A and B are square matrices of the same size, then det(AB) = det(A) det(B) • Example 4 Elementary Linear Algebra

Theorem 2.3.5 • If A is invertible, then Elementary Linear Algebra

2-4 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 Elementary Linear Algebra

2-3 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. Elementary Linear Algebra

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 – Ais singular, which in turn holds if and only if the determinant of I – Aequals zero: det(I – A) = 0 (the so-called characteristic equation of A) Elementary Linear Algebra

2-3 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 = [-tt]T – one eigenvector • If  = 5, x = [3t/4 t]T – the other eigenvector Elementary Linear Algebra

Theorem 2.3.6 (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 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

2 3 4 1 1 3 4 1 2 1 3 4 2 3 4 2 1 4 3 2 4 2 2 3 3 1 2 3 4 1 1 4 1 1 3 4 2 4 2 3 4 3 3 4 4 2 2 3 1 3 1 4 2 3 2 3 1 2 4 4 1 2 1 1 2-4 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} Elementary Linear Algebra

2-4 Inversion • An inversionis said to occur in a permutation (j1, j2, …, jn) 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 j1 and that follow j1 in the permutation; • Find the number of integers that are less than j2 and that follow j2 in the permutation; • Continue the process for j1, j2, …, jn. The sum of these number will be the total number of inversions in the permutation Elementary Linear Algebra

2-4 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 5 + 0 + 1 + 1 + 1 = 8 • The number of inversions is 1 + 2 + 0 = 3 • There no inversions in this permutation Elementary Linear Algebra

2-4 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 Elementary Linear Algebra

2-4 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 Elementary Linear Algebra

2-4 Signed Elementary Product • An nn matrix A has n! elementary products. There are the products of the form a1j1a2j2··· anjn, where (j1, j2, …, jn) is a permutation of the set {1, 2, …, n}. • By a signed elementary product from Awe shall mean an elementary a1j1a2j2··· anjn multiplied by +1 or -1. • We use + if (j1, j2, …, jn) is an even permutation and – if (j1, j2, …, jn) is an odd permutation Elementary Linear Algebra

2-4 Example 6 • List all signed elementary products from the matrices Elementary Linear Algebra

2-4 Determinant • Let A be a square matrix. The determinant functionis 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 Elementary Linear Algebra

2-4 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! Elementary Linear Algebra

2-4 Example 8 • Evaluate the determinants of Elementary Linear Algebra

Elementary Linear Algebra Anton & Rorres, 9 th Edition