Chap. 3 Determinants

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Chap. 3Determinants

• 3.1 The Determinants of a Matrix
• 3.2 Evaluation of a Determinant Using Elementary
  Operations
• 3.3 Properties of Determinants
• 3.4 Introduction to Eigenvalues
• 3.5 Applications of Determinants

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3.1 The Determinant of a Matrix

• Every square matrix can be associated with a real
  number called its determinant.
• Definition The determinant of the matrixis
  given by
• Example 1

?2
2(2) ? 1(?3) 7
2(2) ? 1(4) 0
0(4) ? 2(3) ?6
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Minors and Cofactors of a Matrix
Section 3-1

• If A is a square matrix, then the minor (????)
  Mij of the element aij is the determinant of the
  matrix obtained by deleting the ith row and jth
  column of A.The cofactor (???) Cij is given by
  Cij (?1)ijMij.
• Sign pattern for cofactors

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Theorem 3.1
Section 3-1

• Expansion by Cofactors
• Let A be a square matrix of order n. Then the
  determinant of A is given by
• For any 3?3 matrix

ith row expansion
jth column expansion
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Examples 2 3
Section 3-1

• Find all the minors and cofactors of A, and then
  find the determinant of A.
• Sol

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Example 5
Section 3-1

• Find the determinant of
• Sol

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Example 4
Section 3-1

• Find the determinant of
• Sol Expansion by which row or which column?
• ? the 3rd column three of the entires are zeros

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Triangular Matrices
Section 3-1

• Upper triangular Matrix Lower triangular
  Matrix
• Theorem 3.2 If A is a triangular matrix of order
  n, then its determinant is the product of the
  entires on the main diagonal. That is,

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Example
Section 3-1
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3.2 Evaluation of a Determinant Using
Elementary Operations

• Which of the following two determinants is easier
  to evaluate?

By elementary row operations
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? Theorem 3.3
Section 3-2

• Elementary Row Operations and Determinants
• Let A and B be square matrices.
• 1. If B is obtained from A by interchanging two
  rows of A, then det(B) ?det(A).
• 2. If B is obtained from A by adding a multiple
  of a row of A to another row of A, then det(B)
  det(A).
• 3. If B is obtained from A by multiplying a row
  of A by a nonzero constant c, then det(B)
  cdet(A).

Take a common factor out of a row
? 3
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Example 2
Section 3-2

• Find the determinant of
• Sol

Factor ?7 out of the 2nd row
?(?1)
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Determinants andElementary Column Operations
Section 3-2

• Although Theorem 3.3 was stated in terms of
  elementary row operations, the theorem remains
  valid if the word row is replaced by the word
  column.
• Operations performed on the column of a matrix
  are called elementary column operations.
• Two matrices are called column-equivalent if one
  can be obtained from the other by elementary
  column operations.

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Example 3
Section 3-2

• Find the determinant of
• Sol

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? Theorem 3.4
Section 3-2

• Conditions That Yield a Zero Determinant
• If A is a square matrix and any one of the
  following conditions is true, then det(A) 0.
• 1. An entire row (or an entire column) consists
  of zeros.
• 2. Two rows (or columns) are equal.
• 3. One row (or column) is a multiple of another
  row (or column).

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Examples 4 5
Section 3-2

?(2)
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Example 6
Section 3-2

• Find the determinant of
• Sol

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3.3 Properties of Determinants

• Example 1 Find for the
  matrices
• Sol

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Theorems 3.5 3.6
Section 3-3

• Theorem 3.5 Determinant of a Matrix Product
• If A and B are square matrices of order n,
  thendet(AB) det(A) det(B)
• Remark
• Theorem 3.6 Determinant of a Scalar Multiple of
  a Matrix
• If A is a n?n matrix and c is a scalar, then the
  determinant of cA is given by det(cA) cn
  det(A).
• Remark Thm. 3.3 If B is obtained from A by
  multiplying a row of A by a nonzero constant c,
  then det(B) cdet(A).

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Example 2
Section 3-3

• Find the determinant of the matrix
• Sol

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? Theorems 3.7 3.8
Section 3-3

• Theorem 3.7 Determinant of an Invertible Matrix
• A square matrix A is invertible (nonsingular) if
  and only if det(A) ? 0.
• Theorem 3.8 Determinant of an Inverse Matrix
• If A is invertible, then det(A?1) 1 / det(A).
• Hint A is invertible
• ? AA?1 I

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Example 3 4
Section 3-3

• Example 3 Which of the matrices has an inverse?
• Sol
• Example 4 Find for the matrix
• Sol

It has no inverse.
It has an inverse.
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? Equivalent Conditions for a Nonsingular
Matrix
Section 3-3

• 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
  column vector b.
• 3. Ax O has only the trivial solution.
• 4. A is row-equivalent to In.
• 5. A can be written as the product of elementary
  matrices.
• ? ? Also see in Theorem 2.15 ?
• 6. det(A) ? 0.
• ? See Example 5 (p.148) for instance ?

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Determinant of a Transpose
Section 3-3

• Theorem 3.9 If A is a square matrix, then
  det(A)det(AT).
• Example 6 Show that for the
  following matrix.
• pf

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3.4 Introduction to Eigenvalues

• See Chapter 7

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3.5 Applications of Determinants

• The Adjoint of a MatrixIf A is a square matrix,
  then the matrix of cofactors of A has the form
• The transpose of this matrixis called the
  adjoint of A andis denoted by adj(A).

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Example 1
Section 3-5

• Find the adjoint of
• SolThe matrix of cofactors of A

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? Theorem 3.10
Section 3-5

• The Inverse of a Matrix Given by Its Adjoint
• If A is an n?n invertible matrix, then
• If A is 2?2 matrixthen the adjoint of A is
  .Form Theorem 3.10
  you have

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Example 2
Section 3-5

• Use the adjoint of
  to find .
• Sol

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Theorem 3.11 Cramers Rule
Section 3-5

• If a system of n linear equations in n variables
  has a coefficient matrix with a nonzero
  determinant ,then the solution of the system
  is given bywhere the ith column of Ai is the
  column of constants in the system of equations.

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Example 4
Section 3-5

• Use Cramers Rule to solve the system of linear
  equationfor x.
• Sol

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Area of a Triangle
Section 3-5

• The area of a triangle whose verticesare (x1,
  y1), (x2, y2), and (x3, y3) isgiven bywhere
  the sign (?) is chosen to give a positive area.
• pf Area

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Example 5
Section 3-5

• Fine the area of the triangle whose vertices are
  (1, 0), (2, 2), and (4, 3).
• Sol
• Fine the area of the triangle whosevertices are
  (0, 1), (2, 2), and (4, 3).

(1,0)
Three points in the xy-plane lie on the same line.
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Collinear Pts Line Equation
Section 3-5

• Test for Collinear Points in the xy-PlaneThree
  points (x1, y1), (x2, y2), and (x3, y3) are
  collinearif and only if
• Two-Point Form of the Equation of a LineAn
  equation of the line passing through the distinct
  points (x1, y1) and (x2, y2) is given by

The 3rd point (x, y)
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Example 6
Section 3-5

• Find an equation of the line passing through the
  points(2, 4) and (?1, 3).
• Sol

An equation of the line is x ? 3y ?10.
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Volume of Tetrahedron
Section 3-5

• The volume of the tetrahedron whose vertices are
  (x1,y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4,
  y4, z4), is given by
• where the sign (?) is chosen to give a positive
  area.

• Example 7 Find the volume of the tetrahedron
  whose vertices are (0,4,1), (4,0,0), (3,5,2), and
  (2,2,5).
• Sol

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Coplanar Pts Plane Equation
Section 3-5

• Test for Coplanar Points in SpaceFour points
  (x1,y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4,
  y4, z4) are coplanar if and only if

• Three-Point Form of the Equation of a PlaneAn
  equation of the plane passing through the
  distinct points (x1,y1, z1), (x2, y2, z2), and
  (x3, y3, z3) is given by

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Example 8
Section 3-5

• Find an equation of the plane passing through the
  points(0,1,0), (?1,3,2) and (?2,0,1).
• Sol