Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion


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1 Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates c d a real number with a square matrix. In this chapter, we will learn how to calculate the determinant of n n matrices. Definition: If A is a square matrix of order n, then the minor of entry a ij, denoted M ij, is the determinant of the n 1 n 1 submatrix obtained by removing the ith row and jth column from A. The cofactor of entry a ij is C ij = ( 1) i+j M ij. Example Note: C ij = ±M ij so to find the cofactor you only need to determine the sign once you know the minor entry. 1
2 Theorem 2.1.1: Expansions by cofactors The determinant of a square matrix of order n can be calculated by multiplying any row or column by their cofactors and adding the products. That is, using the cofactor expansion along row i: or along column j: det(a) = a i1 C i1 + a i2 C i2 + + a in C in det(a) = a 1j C 1j + a 2j C 2j + + a nj C nj Note: Since we can choose any row or column for the cofactor expansion, choosing the row or column with the most zeros will cut down on the number of calculations. Note: Choosing any other row or column would give the same answer. 2
3 Definition: If A is a square matrix of order n and C ij is the cofactor of entry a ij, then the matrix of cofactors from A has the form C 11 C 12 C 1n C 21 C 22 C 2n.. C n1 C n2 C nn The transpose of this matrix is called the adjoint of A. That is, adj(a) = = C 11 C 12 C 1n C 21 C 22 C 2n T.. C n1 C n2 C nn C 11 C 21 C n1 C 12 C 22 C n2.. C 1n C 2n C nn 3
4 Theorem 2.1.2: Inverse of a matrix using its adjoint If A is an invertible matrix, then A 1 1 = det(a) adj(a) Definition: A square matrix of order n is upper triangular is all entries below the main diagonal are zero, lower triangular if all entries above the main diagonal are zero, and diagonal is all entries off the main diagonal are zero. Examples: Theorem 2.1.3: If A is a square matrix of order n and is upper triangular, lower triangular or diagonal then det(a) is the product of the entries 4
5 on the main diagonal. That is, Proof: det(a) = a 11 a 22 a nn Examples: 5
6 Theorem 1.7.1: 1. The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. 2. The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular 3. A triangular matrix is invertible if and only if its diagonal entries are all nonzero 4. The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular Proof: See text; uses the adjoint Theorem Cramer s rule: If A x = b is a system of n linear equations and det(a) 0 (that is, A is invertible) then the system has a unique solution given by x 1 = det(a 1) det(a), x 2 = det(a 2) det(a),...,x n = det(a n) det(a) where A j is the matrix obtained by replacing the jth column of A with the vector matrix b. 6
7 Section 2.2: Evaluating determinants by row reduction Theorem 2.2.1: If A is a square matrix of order n and has a row or column of zeros, then det(a) = 0. Proof: Theorem 2.2.2: Let A be a square matrix of order n, then det(a) = det(a T ). Proof: Note: This means almost any statement about determinants that talks about rows is also true for columns. Theorem 2.2.3: Let A be a square matrix of order n 1. If B is the matrix obtained by multiplying a single row or column of A by the scalar k (scaling), then det(b) = k det(a) 2. If B is the matrix obtained by interchanging two rows or columns of A, then det(b) = det(a) 3. If B is the matrix obtained when one row or column of A is added to a multiple of another row or column (replacement), then det(b) = det(a) We can make the same statements about elementary matrices. Theorem 2.2.4: Let E be an elementary square matrix of order n. 1. If E is obtained by a multiplying a row of I n by k, then det(e) = k 2. If E is obtained by interchanging two rows of I n, then det(e) = 1 3. If E results from adding a multiple of one row of I n to another, then det(e) = 1 7
8 Theorem 2.2.5: If A is a square matrix of order n with two proportional rows or columns, then det(a) = 0 Examples: 8
9 There are often ways we can make the calculation of the determinant a little easier. Idea 1: Use elementary row operations to reduce a given matrix to a triangular matrix Note: Be very careful not to use scaling and replacement at the same time (eg replace row1 by row1 + k row2, not by a row1 + b row2) 9
10 Idea 2: Use elementary row operations to produce rows or columns with only one nonzero entry then use cofactor expansion. 10
11 Section 2.3: Properties of the determinant Basic properties of the determinant: 1. det(ka) = k n det(a), k scalar 2. det(a + B) det(a) + det(b) 3. If A, B, C are square matrices of order n that differ only in the rth row, and the rth row of C can obtained by adding the corresponding entries in the rth row if A and B, then det(c) = det(a) + det(b) (Theorem 2.3.1) 4. If E is an elementary matrix, then det(eb) = det(e)det(b) (Lemma 2.3.2) 5. A square matrix is invertible if and only if det(a) 0 (Theorem 2.3.3) 6. If A and B are square matrices of order n, then det(ab) = det(a)det(b) (Theorem 2.3.4) 7. If A is invertible, then det(a 1 ) = 1 det(a) Examples: (Theorem 2.3.5) 11
12 Theorem 2.3.6: If A is a square matrix of order n, then the following statements are equivalent 1. A is invertible 2. A x = 0 has only the trivial solution 3. The reduced row echelon form of A is I n 4. A can be expressed as a product of elementary matrices 5. A x = b is consistent for every n 1 matrix b 6. A x = b has exactly one solution for every n 1 matrix b 7. det(a) 0 Often we need to solve problems of the form A x = λ x where λ is a scalar. This is a system of n linear equations. Definition: The value(s) of λ for which the system of equations (λi A) x = 0 has a nontrivial solution of called the characteristic value or eigenvalue of A. If λ is an eigenvalue of A, then the nontrivial solutions of the system are called the eigenvectors of A corresponding to λ. Note: Eigenvalues and eigenvectors are used alot, especially when modelling and analyzing physical systems. 12
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