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MAT%202401%20Linear%20Algebra

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MAT 2401 Linear Algebra 3.1 The Determinant of a Matrix http://myhome.spu.edu/lauw HW Written Homework Preview How do I know a matrix is invertible? – PowerPoint PPT presentation

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Title: MAT%202401%20Linear%20Algebra


1
MAT 2401Linear Algebra
  • 3.1 The Determinant of a Matrix

http//myhome.spu.edu/lauw
2
HW
  • Written Homework

3
Preview
  • How do I know a matrix is invertible?
  • We will look at determinant that tells us the
    answer.

4
Recall
Therefore, if D?0, D is called the _________ of A
  • If Dad-bc ? 0 the inverse of
  • is given by

5
Fact
If D0, A is singular. To see this, for a ? 0, we can do the following
  • If Dad-bc 0 the inverse of
  • DNE.

6
The Task
  • Given a square matrix A, we wish to associate
    with A a scalar det(A) that will tell us whether
    or not A is invertible

7
Fact (3.3)
  • A square matrix A is invertible
  • if and only if det(A)?0

8
Interesting Comments
  • Interesting comments from a text
  • The concept of determinant is subtle and not
    intuitive, and researchers had to accumulate a
    large body of experience before they were able to
    formulate a correct definition for this number.

9
n2
1. Notations 2. Mental picture for memorizing
10
n3
11
n3
Q1 What? Do I need to remember this? Q2 What if A is 4x4 or bigger? Q3 Is there a formula for 1x1 matrix?
12
Observations
13
Observations
14
Observations
15
Observations
We need 1. a notion of one size smaller but related determinants. 2. a way to assign the correct signs to these smaller determinants. 3. a way to extend the computations to nxn matrices.
16
Minors and Cofactors
Example
  • Aaij, a nxn Matrix.
  • Let Mij be the determinant of the
  • (n-1)x(n-1) matrix obtained from A by deleting
    the row and column containing aij.
  • Mij is called the minor of aij.


17
Minors and Cofactors
Example
  • Aaij, a nxn Matrix.
  • Let Cij (-1)ij Mij
  • Cij is called the cofactor of aij.

18
n3

19
Determinants
  • Formally defined Inductively by using cofactors
    (minors) for all nxn matrices in a similar
    fashion.
  • The process is sometimes referred as Cofactors
    Expansion.

20
Cofactors Expansion (across the first column)
  • The determinant of a nxn matrix Aaij is a
    scalar defined by

21
Example 1
22
Remark
  • The cofactor expansion can be done across any
    column or any row.


23
Sign Pattern
24
Cofactors Expansion
25
Special Matrices and Their Determinants
  • (Square) Zero Matrix
  • det(O)?
  • Identity Matrix
  • det(I)?
  • We will come back to this later.

26
Upper Triangular Matrix
27
Lower Triangular Matrix
28
Diagonal Matrix
29
Diagonal Matrix
  • Q T or F A diagonal matrix
  • is upper triangular?

30
Example 2
31
Determinant of a Triangular Matrix
  • Let Aaij, be a nxn Triangular Matrix,
  • det(A)

32
Special Matrices and Their Determinants
  • (Square) Zero Matrix
  • det(O)
  • Identity Matrix
  • det(I)
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