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3 - 1 Chapter 2B Determinants 2B.1 The Determinant and Evaluation of a Matrix 2B.2 Properties of Determinants 2B.3 Eigenvalues and Application of Determinants 2B.4 Geometry of Determinants: Determinants as Size Functions When we look at a particular square matrix, the question of whether it is nonsingular is one of the first things that we ask. This chapter develops a formula to determine this.

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3 - 2 2B.1 The Determinant of a Matrix The determinant of a 2 × 2 matrix: Note:

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3 - 3 Minor of the entry : Cofactor of :

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3 - 4 Ex: Notes: Sign pattern for cofactors

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3 - 5 Thm 3B.1: Thm 3B.1: (Expansion by cofactors) Cofactor expansion along the i-th row (Cofactor expansion along the i-th row, i=1, 2,…, n ) Cofactor expansion along the j-th column (Cofactor expansion along the j-th column, j=1, 2,…, n ) Let A is a square matrix of order n, then the determinant of A is given by or

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3 - 6 Ex: The determinant of a matrix of order 3

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3 - 7 Ex 5: (The determinant of a matrix of order 3) Sol:

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3 - 8 The determinant of a matrix of order 3: –4 0 16 –12 06 Ex : Ex :

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3 - 9 Upper triangular Upper triangular matrix: Lower triangular Lower triangular matrix: Diagonal matrix: below All the entries below the main diagonal are zeros. above All the entries above the main diagonal are zeros. All the entries above and below the main diagonal are zeros. upper triangularlower triangular diagonal

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3 - 10 Theorem 2B.2: Theorem 2B.2: Determinant of a Triangular Matrix If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is At this moment, our primary way to decide whether a matrix is singular is to do Gaussian reduction and then check whether the diagonal of resulting echelon form matrix has any zeroes. We will look for a family of functions with the property of being unaffected by row operations and with the property that a determinant of an echelon form matrix is the product of its diagonal entries.

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3 - 11 Ex: Ex: Find the determinants of the following triangular matrices. (a)(a) (b)(b) |A| = (2)(–2)(1)(3) = –12 |B| = (–1)(3)(2)(4)(–2) = 48 (a)(a) (b)(b) Sol:

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3 - 12 Keywords in This Section: determinant : 行列式 minor : 子行列式 cofactor : 餘因子 expansion by cofactors : 餘因子展開 upper triangular matrix: 上三角矩陣 lower triangular matrix: 下三角矩陣 diagonal matrix: 對角矩陣

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3 - 13 2B.2 Evaluation of a determinant using elementary operations Theorem 2B.3: Theorem 2B.3: Elementary row operations and determinants Let A and B be square matrices,

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3 - 14 Ex:

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3 - 15 Note: A row-echelon form of a square matrix is always upper triangular. Ex: Ex: Evaluation a determinant using elementary row operationsSol:

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3 - 17 Theorem 2B.4: Theorem 2B.4: Conditions that yield a zero determinant (a) An entire row (or an entire column) consists of zeros. (b) Two rows (or two columns) are equal. (c) One row (or column) is a multiple of another row (or column). If A is a square matrix and any of the following conditions is true, then det (A) = 0. The theorem states that : a matrix with two identical rows or two linear dependent rows has a determinant of zero. A matrix with a zero row has a determinant of zero. Note that a matrix is nonsingular if and only if its determinant is nonzero and the determinant of an echelon form matrix is the product down its diagonal. Do Gaussian reduction, keeping track of any changes of sign caused by row swaps and any scalars that are factored out, and then finish by multiplying down the diagonal of the echelon form result This theorem provides a way to compute the value of a determinant function on a matrix: Do Gaussian reduction, keeping track of any changes of sign caused by row swaps and any scalars that are factored out, and then finish by multiplying down the diagonal of the echelon form result.

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3 - 18 Cofactor ExpansionRow Reduction Order n AdditionsMultiplications Additions Multiplications 359510 51192053045 103,628,7996,235,300285339 Note:

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3 - 19 Ex: (Evaluating a determinant) Sol:

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3 - 21 2B.2 Properties of Determinants Notes: Theorem 2B.5: Theorem 2B.5: Determinant of a matrix product (1) det (EA) = det (E) det (A) (2) (3) det (AB) = det (A) det (B)

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3 - 22 Ex: (The determinant of a matrix product) Sol: Find |A|, |B|, and |AB|

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3 - 24 Ex: Find |A|. Sol: Theorem 2B.6 Theorem 2B.6: Determinant of a scalar multiple of a matrix If A is an n × n matrix and c is a scalar, then cc n det (cA) = c n det (A)

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3 - 25 Ex: (Classifying square matrices as singular or nonsingular) A has no inverse (it is singular). B has inverse (it is nonsingular). Sol: Thm 2B.7 Thm 2B.7: Determinant of an invertible matrix A square matrix A is invertible (nonsingular) if and only if det (A) 0

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3 - 26 Ex: (a) (b) Sol: Thm 2B.8 Thm 2B.8: Determinant of an inverse matrix Thm 2B.9 Thm 2B.9: Determinant of a transpose

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3 - 27 If A is an n × n matrix, then the following statements are equivalent. (1) A is invertible. (2) Ax = b has a unique solution for every n × 1 matrix b. (3) Ax = 0 has only the trivial solution of zero column vector. (4) A is row-equivalent to I n (5) A can be written as the product of elementary matrices. det (A) 0 (6) det (A) 0 nonsingular matrix Equivalent conditions for a nonsingular matrix:

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3 - 28 Ex: Which of the following system has a unique solution? (a)(a) This system does not have a unique solution. Sol:

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3 - 29 Sol: (b)(b) This system has a unique solution.

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3 - 30 2B.3 Introduction to Eigenvalues Eigenvalue problem: If A is an n n matrix, do there exist nonzero n 1 matrices x such that Ax is a scalar multiple of x ？ Eigenvalue and eigenvector: A ： an n n matrix ： a scalar x ： a n 1 nonzero column matrix Eigenvalue Eigenvector (The fundamental equation for the eigenvalue problem)

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3 - 31 Ex 1: (Verifying eigenvalues and eigenvectors) Eigenvalue Eigenvector

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3 - 32 Question: Given an n n matrix A, how can you find the eigenvalues and corresponding eigenvectors? Characteristic equation Characteristic equation of A M n n : Note: If has nonzero solutions iff. Note: (homogeneous system)

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3 - 33 Ex: (Finding eigenvalues and eigenvectors) Sol: Characteristic equation: Eigenvalue:

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3 - 35 Application Example of Eigenvalue-Eigenvector Problem The equations of motion for identical mass and spring constant can be described by Try the solution and plug this into the differential equations: We can obtain Rearrange these to put them into a neater form

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3 - 36 A nontrivial solution occurs when the determinant is zero, which yields the following solutions (eigenvalues): With the given eigenvalues, we can find the corresponding eigenvectors (normal modes) to be

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3 - 37 2.3 Applications of Determinants Matrix of cofactors of A: Adjoint matrix Adjoint matrix of A:

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3 - 38 Thm 2B.10 Thm 2B.10 : The inverse of a matrix given by its adjoint If A is an n × n invertible matrix, then Ex:

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3 - 39 Ex: (a) Find the adjoint of A. (b) Use the adjoint of A to find Sol:

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3 - 40 cofactor matrix of A adjoint matrix of A inverse matrix of A Check:

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3 - 41 Thm 2B.11 : Thm 2B.11 : Cramer’s Rule (this system has a unique solution)

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3 - 42 ( i.e.,)

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3 - 43 Pf: A x = b,

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3 - 45 Ex: Use Cramer’s rule to solve the system of linear equations. Sol:

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3 - 46 Keywords in This Section: matrix of cofactors : 餘因子矩陣 adjoint matrix : 伴隨矩陣 Cramer’s rule : Cramer 法則

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3 - 47 2B.4 Geometry of Determinants: Determinants as Size Functions We have so far only considered whether or not a determinant is zero, here we shall give a meaning to the value of that determinant. One way to compute the area that it encloses is to draw this rectangle and subtract the area of each subregion. O

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3 - 48 The properties in the definition of determinants make reasonable postulates for a function that measures the size of the region enclosed by the vectors in the matrix. See this case: The region formed by and is bigger, by a factor of k, than the shaded region enclosed by and. That is, size (, ) = k · size(, ).

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3 - 49 pivoting Another property of determinants is that they are unaffected by pivoting. Here are before-pivoting and after-pivoting boxes (the scalar used is k = 0.35). Although the region on the right, the box formed by and, is more slanted than the shaded region, the two have the same base and the same height and hence the same area. This illustrates that

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3 - 50 That is, we’ve got an intuitive justification to interpret det (,..., ) as the size of the box formed by the vectors. Example Example The volume of this parallelepiped, which can be found by the usual formula from high school geometry, is 12.

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3 - 51 counterclockwisepositive clockwisenegative The only difference between them is in the order in which the vectors are taken. If we take first and then go to, follow the counterclockwise are shown, then the sign is positive. Following a clockwise are gives a negative sign. The sign returned by the size function reflects the ‘orientation’ or ‘sense’ of the box.

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3 - 52 Volume, because it is an absolute value, does not depend on the order in which the vectors are given. The volume of the parallelepiped in the following example, can also be computed as the absolute value of this determinant. The definition of volume gives a geometric interpretation to something in the space, boxes made from vectors.

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3 - 53 t Application of the map t represented with respect to the standard bases by will double sizes of boxes, e.g., from this to

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