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1 4. Determinants

2 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant

3 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant

4 4.1. Determinants; Cofactor Expansion Elementary Products (4) Elementary product: a product containing one entry from each row and one entry from each column. In formula (4) each elementary product is of the form where the blank contain some permutation of the column indices {1, 2, 3}. Signed elementary product The sign can be determined by counting the minimum number of interchanges in the permutation of the column indices required to put those indices into their natural order: the sign is + if the number is even and if it is odd.

5 4.1. Determinants; Cofactor Expansion Elementary Products

6 4.1. Determinants; Cofactor Expansion General Determinants n n determinant or n-th order determinant The signed elementary products are to be summed over all possible permutations {j 1, j 2,, j n } of the column indices.

7 4.1. Determinants; Cofactor Expansion Determinants of Matrices with Rows or Columns That Have All Zeros Determinants of Triangular Matrices

8 4.1. Determinants; Cofactor Expansion Minors and Cofactors

9 4.1. Determinants; Cofactor Expansion Minors and Cofactors Example 3

10 4.1. Determinants; Cofactor Expansion Cofactor Expansions These are called cofactor expansions of A. Note that in each cofactor expansion, the entries and cofactors all come from the same row or the same column.

11 4.1. Determinants; Cofactor Expansion Cofactor Expansions

12 4.1. Determinants; Cofactor Expansion Cofactor Expansions Example 5 Use a cofactor expansion to find the determinant of

13 4.2. Properties of Determinants Determinant of A T

14 4.2. Properties of Determinants Effect of Elementary Row Operations on A Determinant

15 4.2. Properties of Determinants Effect of Elementary Row Operations on A Determinant (a) Suppose that the ith row of A is multiplied by the scalar k to produce the matrix B, Since the ith row is deleted when the cofactors along that row are computed, the cofactors in this formula are unchanged when the ith row is multiplied by k.

16 4.2. Properties of Determinants Effect of Elementary Row Operations on A Determinant Example 1

17 4.2. Properties of Determinants Effect of Elementary Row Operations on A Determinant Example 2

18 4.2. Properties of Determinants Effect of Elementary Row Operations on A Determinant Example 3 What is the relationship between det(a) and det(-a)? Thus, det(-a)=det(a) if n is even, and det(-a)=-det(a) if n is odd.

19 4.2. Properties of Determinants Simplifying Cofactor Expansions (1) A cofactor expansion can be minimized by expanding along a row or column with the maximum number of zeros. (2) Adding multiples of one row (or column) to another does not change the determinant of the matrix. Example 4 Use a cofactor expansion to find the determinant of

20 4.2. Properties of Determinants A Determinant Test for Invertibility If R is the reduced row echelon form of square matrix A, determinant of the matrices are both zero or both nonzero. Assume that A is invertible, in which case the reduced row echelon form of A is I (by Theorem 3.3.3). Since det(i) 0, it follows that det(a) 0. Conversely, assume that det(a) 0. Since det(r) 0, R=I. Thus, A is invertible (by Theorem 3.3.3)

21 4.2. Properties of Determinants Determinants of A Product of Matrices Determinants of The Inverse of A Matrix Since AA -1 =I, det(aa -1 )=det(i)=1=det(a)det(a -1 ). Thus, det(a -1 )=1/det(A).

22 4.2. Properties of Determinants Determinants of A+B det(a+b) det(a)+det(b) Example 8 det(a)=1, det(b)=5, det(a+b)=23.

23 4.2. Properties of Determinants Determinant Evaluation by LU-Decomposition If A=LU, then det(a)=det(l)det(u), which is easy to compute since L and U are triangular. Thus, nearly all of the computational work in evaluating det(a) is expended in obtaining the LU-decomposition. From Table The number of flops required to obtain the LU-decomposition of an n n matrix is on the order of 2/3n 3 for large values of n. This is an enormous improvement over the determinant definition, which involves the computation of n! signed elementary products. For example, today s typical PC can evaluate determinant in less than one-thousandth of a second by LU-decomposition compared to the roughly years that would be required for it to evaluate 30! Signed elementary products.

24 4.2. Properties of Determinants A Unifying Theorem

25 4.3. Cramer s s Rule Adjoint of A Matrix

26 4.3. Cramer s s Rule Adjoint of A Matrix If we multiply the entries in the first row by the corresponding cofactors from the third row, then the sum is Let s consider a matrix A that results when the third row of A is replaced by a duplicate of the first row. We know that det(a )=0 because of the duplicate rows.

27 4.3. Cramer s s Rule Adjoint of A Matrix

28 4.3. Cramer s s Rule Adjoint of A Matrix Example 1 Find the matrix of cofactors from A.

29 4.3. Cramer s s Rule A Fomula for The Inverse of A Matrix Suppose that A is invertible. The entry in the ith row and jth column of this product is (3)

30 4.3. Cramer s s Rule A Fomula for The Inverse of A Matrix In the case where i=j, the entries and cofactors come from the same row of A, so (3) is the cofactor expansion of det(a) along that row. In the case where i j, the entries and cofactors come from different rows, so the sum is zero by Theorem Since A is invertible, it follows that det(a) 0, so this equation can be rewritten as

31 4.3. Cramer s s Rule A Fomula for The Inverse of A Matrix Example 2 Find the inverse of the matrix A in Example 1.

32 4.3. Cramer s s Rule How The Inverse Formula Is Used Computer program usually use LU-decomposition (as discussed in Section 3.7) and not Formula (2) to invert matrices. Thus, the value of Formula (2) is not for numerical computations, but rather as a tool in theoretical analysis.

33 4.3. Cramer s s Rule Cramer s Rule

34 4.3. Cramer s s Rule Cramer s Rule In the case where det(a) 0, we can use Formula (2) to rewrite the unique solution of Ax=b as Therefore, the entry in the jth row of x is The cofactors in this expression come from the jth column of A and hence remain unchanged if we replace the jth column of A by b (the jth column is crossed out when the cofactors are computed). Since this substitution produces the matrix A j, the numerator in (5) can be interpreted as the cofactor expansion along the jth column of A j. Thus,

35 4.3. Cramer s s Rule Cramer s Rule Example 4 Use Cramer s rule to solve the system

36 4.3. Cramer s s Rule Cramer s Rule Example 5 Use Cramer s rule to solve the system

37 4.3. Cramer s s Rule Geometric Interpretation of Determinants

38 4.3. Cramer s s Rule Geometric Interpretation of Determinants (a) Suppose that the matrix A is partitioned into columns as and let us assume that the parallelogram with adjacent sides u and v is not degenerate. We can see that this area can be expressed as

39 4.3. Cramer s s Rule Geometric Interpretation of Determinants The square of the area can be expressed as (area) 2 =

40 4.3. Cramer s s Rule Geometric Interpretation of Determinants

41 4.3. Cramer s s Rule Geometric Interpretation of Determinants Example 8 Find the area of the triangle with vertices A(-5, 4), B(3, 2), and C(-2, -3).

42 4.3. Cramer s s Rule Cross Products Standard unit vector i=(1,0,0), j=(0,1,0), k=(0,0,1)

43 4.3. Cramer s s Rule Cross Products Example 9 Let u=(1,2,-2) and v=(3,0,1). Find (a) u v (b) v u (c) u u

44 4.3. Cramer s s Rule Cross Products

45 4.3. Cramer s s Rule Cross Products

46 4.3. Cramer s s Rule Cross Products In general, if u and v are nonzero and nonparallel vectors, then the direction of u v in relation to u and v can be determined by the right-hand rule.

47 4.3. Cramer s s Rule Cross Products The associative law does not hold for cross products; for example,

48 4.3. Cramer s s Rule Cross Products

49 4.3. Cramer s s Rule Cross Products (a) Since 0 θ π, it follows that sin θ 0 and hence that Thus,

50 4.3. Cramer s s Rule Cross Products (b) Example 10 Find the area of the triangle in R 3 that has vertices P 1 (2,2,0), P 2 (-1,0,2), and P 3 (0,4,3).

51 4.4. A First Look at Eigenvalues and Eigenvectors Fixed Points Recall that a fixed point of an n n matrix A is a vector x in R n such that Ax=x (see the discussion preceding Example 6 of Section 3.6) Every square matrix A has at least one fixed point, namely x=0. We call this the trivial fixed point of A. Ax=x Ax=Ix (A-I)x=0

52 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors One might also consider more general equations of the form Ax= λx in which λ is a scalar.

53 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors The most direct way of finding the eigenvalues of an n n matrix A is to rewrite the equation Ax= λx as Ax= λix, or equivalently, as (4) and then try to determine those values of λ, if any, for which this system has nontrivial solutions. Since (4) has nontrivial solutions if and only if the coefficient matrix λi-a is singular, we see that the eigenvalues of A are the solutions of the equation This is called the characteristic equation of A. Also, if λ is an eigenvalue of A, then (4) has a nonzero solution space, which we call the eigenspace of A corresponding to λ. It is the nonzero vectors in the eigenspace of A corresponding to λ that are the eigenvectors of A corresponding to λ.

54 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors Example 2 (a) Find the eigenvalues and corresponding eigenvectors of the matrix (b) Graph the eigenspaces of A in an xy-coordiante system.

55 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalues of Triangular Matrices If A is an n n triangular matrix with diagonal entries a 11, a 22,, a nn, then λi-a is a triangular matrix with diagonal entries λ-a 11, λ-a 22,, λ-a nn. Thus the characteristic polynomial of A is which implies that the eigenvalues of A are

56 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalues of Powers of A Matrix If λ is an eigenvalue of A and x is a corresponding eigenvector, then which shows that λ 2 is an eigenvalue of A 2 and x is a corresponding eigenvector.

57 4.4. A First Look at Eigenvalues and Eigenvectors A Unifying Theorem λ=0 is an eigenvalue of A if and only if there is a nonzero vector x such that Ax=0.

58 4.4. A First Look at Eigenvalues and Eigenvectors Algebraic Multiplicity REMARK This theorem implies that an n n matrix has n eigenvalues if we agree to count repetitions and allow complex eigenvalues, but the number of distinct eigenvalues may be less than n.

59 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalue Analysis of 2 2 Matrices

60 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalue Analysis of 2 2 Symmetric Matrices Later in the text we will show that all symmetric matrices with real entries have real eigenvalues.

61 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalue Analysis of 2 2 Symmetric Matrices If the 2 2 symmetric matrix is then so Theorem implies that A has real eigenvalues. It also follows from that theorem that A has one repeated eigenvalue if and only if Since this holds if and only if a=d and b=0.

62 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalue Analysis of 2 2 Symmetric Matrices Example 6 Graph the eigenspaces of the symmetric matrix in an xy-coordinate system.

63 4.4. A First Look at Eigenvalues and Eigenvectors Expressions for Determinant and Trace in Terms of Eigenvalues (a) Write the characteristic polynomial in factored form: Setting λ=0 yields But det(-a)=(-1) n det(a), so it follows that

64 4.4. A First Look at Eigenvalues and Eigenvectors Expressions for Determinant and Trace in Terms of Eigenvalues (b) Assume that A=[a ij ], so we can write p(λ) as (30) Any elementary product that contains an entry that is off the main diagonal of (30) as a factor will contain at most n-2 factors that involve λ. Thus the coefficient of λ n-1 in p(λ) is the same as the coefficient of λ n-1 in the product Expanding this product we see that it has the form

65 4.4. A First Look at Eigenvalues and Eigenvectors Expressions for Determinant and Trace in Terms of Eigenvalues and expanding the expression for p(λ) we see that it has the form = Thus, we must have

66 4.4. A First Look at Eigenvalues and Eigenvectors Expressions for Determinant and Trace in Terms of Eigenvalues Example 7 Find the determinant and trace of a 3 3 matrix whose characteristic polynomial is This polynomial can be factored as

67 4.4. A First Look at Eigenvalues and Eigenvectors Eigenvalues by Numerical Methods Eigenvalues are rarely obtained by solving the characteristic equation in real-world applications primarily for two reasons: 1. In order to construct the characteristic equation of an n n matrix A, it is necessary to expand the determinant det(λi-a). The computations are prohibitive for matrices of the size that occur in typical applications. 2. There is no algebraic formula of finite algorithm that can be used to obtain the exact solutions of the characteristic equation of a general n n matrix when n 5. Given these impediments, various algorithms have been developed for producing numerical approximations to the eigenvalues and eigenvectors.

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