3 - 1 Chapter 2B Determinants 2B.1 The Determinant and Evaluation of a Matrix 2B.2 Properties of...
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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
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.
Slide 2
3 - 2 2B.1 The Determinant of a Matrix The determinant of a 2 2
matrix: Note:
Slide 3
3 - 3 Minor of the entry : Cofactor of :
Slide 4
3 - 4 Ex: Notes: Sign pattern for cofactors
Slide 5
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
Slide 6
3 - 6 Ex: The determinant of a matrix of order 3
Slide 7
3 - 7 Ex 5: (The determinant of a matrix of order 3) Sol:
Slide 8
3 - 8 The determinant of a matrix of order 3: 4 0 16 12 06 Ex :
Ex :
Slide 9
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
Slide 10
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.
Slide 11
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:
Slide 12
3 - 12 Keywords in This Section: determinant : minor : cofactor
: expansion by cofactors : upper triangular matrix: lower
triangular matrix: diagonal matrix:
Slide 13
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,
Slide 14
3 - 14 Ex:
Slide 15
3 - 15 Note: A row-echelon form of a square matrix is always
upper triangular. Ex: Ex: Evaluation a determinant using elementary
row operationsSol:
Slide 16
3 - 16
Slide 17
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.
Slide 18
3 - 18 Cofactor ExpansionRow Reduction Order n
AdditionsMultiplications Additions Multiplications 359510
51192053045 103,628,7996,235,300285339 Note:
Slide 19
3 - 19 Ex: (Evaluating a determinant) Sol:
Slide 20
3 - 20
Slide 21
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)
Slide 22
3 - 22 Ex: (The determinant of a matrix product) Sol: Find |A|,
|B|, and |AB|
Slide 23
3 - 23
Slide 24
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)
Slide 25
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
Slide 26
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
Slide 27
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:
Slide 28
3 - 28 Ex: Which of the following system has a unique solution?
(a)(a) This system does not have a unique solution. Sol:
Slide 29
3 - 29 Sol: (b)(b) This system has a unique solution.
Slide 30
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)
Slide 31
3 - 31 Ex 1: (Verifying eigenvalues and eigenvectors)
Eigenvalue Eigenvector
Slide 32
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)
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
Slide 36
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
Slide 37
3 - 37 2.3 Applications of Determinants Matrix of cofactors of
A: Adjoint matrix Adjoint matrix of A:
Slide 38
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:
Slide 39
3 - 39 Ex: (a) Find the adjoint of A. (b) Use the adjoint of A
to find Sol:
Slide 40
3 - 40 cofactor matrix of A adjoint matrix of A inverse matrix
of A Check:
Slide 41
3 - 41 Thm 2B.11 : Thm 2B.11 : Cramers Rule (this system has a
unique solution)
Slide 42
3 - 42 ( i.e.,)
Slide 43
3 - 43 Pf: A x = b,
Slide 44
3 - 44
Slide 45
3 - 45 Ex: Use Cramers rule to solve the system of linear
equations. Sol:
Slide 46
3 - 46 Keywords in This Section: matrix of cofactors : adjoint
matrix : Cramers rule : Cramer
Slide 47
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
Slide 48
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(, ).
Slide 49
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
Slide 50
3 - 50 That is, weve 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.
Slide 51
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.
Slide 52
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.
Slide 53
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