MAT%202401%20Linear%20Algebra

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MAT 2401Linear Algebra

• 3.1 The Determinant of a Matrix

http//myhome.spu.edu/lauw
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HW

• Written Homework

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Preview

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

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Recall
Therefore, if D?0, D is called the _________ of A

• If Dad-bc ? 0 the inverse of
• is given by

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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.

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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

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Fact (3.3)

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

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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.

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n2
1. Notations 2. Mental picture for memorizing
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n3
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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?
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Observations
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Observations
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Observations
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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.
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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.

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Minors and Cofactors
Example

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

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n3

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Determinants

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

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Cofactors Expansion (across the first column)

• The determinant of a nxn matrix Aaij is a
  scalar defined by

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Example 1
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Remark

• The cofactor expansion can be done across any
  column or any row.

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Sign Pattern
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Cofactors Expansion
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Special Matrices and Their Determinants

• (Square) Zero Matrix
• det(O)?
• Identity Matrix
• det(I)?
• We will come back to this later.

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Upper Triangular Matrix
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Lower Triangular Matrix
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Diagonal Matrix
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Diagonal Matrix

• Q T or F A diagonal matrix
• is upper triangular?

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Example 2
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Determinant of a Triangular Matrix

• Let Aaij, be a nxn Triangular Matrix,
• det(A)

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Special Matrices and Their Determinants

• (Square) Zero Matrix
• det(O)
• Identity Matrix
• det(I)