Elementary Linear Algebra

1
Elementary Linear Algebra

• Determinants

2
Chapter Content

• Determinants by Cofactor Expansion
• Evaluating Determinants by Row Reduction
• Properties of the Determinant Function
• A Combinatorial Approach to Determinants

3
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)ij Mij

4
Minor and Cofactor

• Remark
• Note that Cij ?Mij and the signs (-1)ij in the
  definition of cofactor form a checkerboard
  pattern

5
Example

• Let
• The minor of entry a11 is
• The cofactor of a11 is C11 (-1)11M11 M11
  16

6
Example

• Similarly, the minor of entry a32 is
• The cofactor of a32 is C32 (-1)32M32 -M32
  -26

7
Cofactor Expansion

• 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)

8
Cofactor Expansion

• Example

9
Adjoint of a Matrix

• Definition
• 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)

10
Adjoint of a Matrix

• Remarks
• In a cofactor expansion we compute det(A) by
  multiplying the entries in a row or column by
  their cofactors and adding the resulting
  products.
• If one multiplies the entries in any row by the
  corresponding cofactors from a different row, the
  sum of these products is always zero.

11
Example

• 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

12
Theorems

• Theorem 2.1.2 (Inverse of a Matrix using its
  Adjoint)
• If A is an invertible matrix, then

13
Example

• The inverse (see below) is

14
Theorems

• Theorem 2.1.4 (Cramers 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 jth column of A by the entries in
  the matrix b b1 b2 bnT

15
Example

• Use Cramers rule to solve
• Since
• Thus,

16
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.
• det(A) det(AT)

17
Theorems

• Theorem 2.2.2 (Triangular Matrix)
• 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 that is, det(A)
  a11a22ann

18
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)

19
Theorems

• 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

20
Theorems

• 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

21
Example (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
22
Example

-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.
23
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)

24
Basic Properties of Determinant

• There is no simple relationship exists between
  det(A), det(B), and det(AB) in general.
• In particular, we emphasize that det(AB) is
  usually not equal to det(A) det(B).

25
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.
• Example

26
Theorems

• 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)

27
Theorems

• Theorem 2.3.3 (Determinant Test for
  Invertibility)
• A square matrix A is invertible if and only if
  det(A) ? 0

28
Theorems

• Theorem 2.3.4
• If A and B are square matrices of the same size,
  then
• det(AB) det(A) det(B)
• Theorem 2.3.5
• If A is invertible, then

29
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
• For example

30
Linear Systems of the Form Ax ?x

• Such systems are really homogeneous linear in
  disguise, since the expresses can be rewritten as
  (?I A)x 0

31
Eigenvalue and Eigenvector

• Definition
• 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.
• 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.

32
Eigenvalue and Eigenvector

• Theorem (Eigenvalues and Singularity)
• ? is an eigenvalue of A if and only if ?I A is
  singular, which in turn holds if and only if the
  determinant of ?I A equals zero
• det(?I A) 0
• (the so-called characteristic equation of A)

33
Example

• The linear system
• The characteristic equation of A is
• The eigenvalues of A are ? -2 and ? 5

34
Example

• 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 -t tT one eigenvector
• If ? 5, x 3t/4 tT the other eigenvector

35
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

36
Permutation(????)

• Definition
• A permutation of the set of integers 1,2,,n is
  an arrangement of these integers in some order
  without omission repetition

37
Permutation

• Example
• There are six different permutations of the set
  of integers 1,2,3.
• These are(1,2,3), (2,1,3), (3,1,2), (1,3,2),
  (2,3,1), (3,2,1).
• Example
• List all permutations of the set of integers
  1,2,3,4

38
Inversion (??)

• Definition
• An inversion is said to occur in a permutation
  (j1, j2, , jn) whenever a larger integer
  precedes a smaller one.

39
Inversion

• Remark
• 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

40
Example

• 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

41
Permutation

• Definition
• 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

42
Permutation

• Example
• The following table classifies the various
  permutations of 1,2,3 as even or odd

43
Elementary Product

• Definition
• 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.

44
Elementary Product

• Example
• The elementary product of the matrix is

45
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 A we 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

46
Example

• List all signed elementary products from the
  matrices

47
Determinant

• Definition
• Let A be a square matrix. The determinant
  function is 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

48
Determinant

• Example

49
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!

-
50
Notation and Terminology

• We note that the symbol A is an alternative
  notation for det(A)
• The determinant of A is often written
  symbolically a
• det(A) ? ? a1j1a2j2 anjn
• where ? indicates that the terms are to be
  summed over all permutations (j1, j2, , jn) and
  the or is selected in each term according to
  where the permutation is even or odd

51
Notation and Terminology

• Remark
• Evaluating determinants directly from the above
  definition leads to computational difficult.
• 4?4 matrices need 4! 24 signed elementary
  products
• 10?10 determinant need 10! 3628800 signed
  elementary products!
• Other methods are required.