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MAT 2401Linear 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?
- We will look at determinant that tells us the

answer.

Recall

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

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

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.

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

Fact (3.3)

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

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.

n2

1. Notations 2. Mental picture for memorizing

n3

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?

Observations

Observations

Observations

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.

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.

Minors and Cofactors

Example

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

n3

Determinants

- Formally defined Inductively by using cofactors

(minors) for all nxn matrices in a similar

fashion. - The process is sometimes referred as Cofactors

Expansion.

Cofactors Expansion (across the first column)

- The determinant of a nxn matrix Aaij is a

scalar defined by

Example 1

Remark

- The cofactor expansion can be done across any

column or any row.

Sign Pattern

Cofactors Expansion

Special Matrices and Their Determinants

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

Upper Triangular Matrix

Lower Triangular Matrix

Diagonal Matrix

Diagonal Matrix

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

Example 2

Determinant of a Triangular Matrix

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

Special Matrices and Their Determinants

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

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