BAI

CM20144

Applications I:

Mathematics for Applications

Mark Wood

cspmaw@cs.bath.ac.uk

http://www.cs.bath.ac.uk/~cspmaw

BAI

• Determinants

Evaluation Methods

Properties

Examples

• Test 5

Today’s Tutorial

BAIEvaluating Determinants 1

BAI

• ‘Diagonals’ Method

Only works for 2 x 2 and 3 x 3

Multiply forward diagonal elements and add

Multiply backward diagonal elements and subtract

Evaluating Determinants 1

BAI

• ‘Diagonals’ Method

Only works for 2 x 2 and 3 x 3

Multiply forward diagonal elements and add

Multiply backward diagonal elements and subtract

• Cofactor Method

Pick the row or column with the most zeros

Calculate the cofactor for each element and sum

Cofactor = sign x minor

Signs alternate

Minor = determinant of remaining matrix…

Evaluating Determinants 1

BAI

• ‘Diagonals’ Method

Only works for 2 x 2 and 3 x 3

Multiply forward diagonal elements and add

Multiply backward diagonal elements and subtract

• Cofactor Method

Pick the row or column with the most zeros

Calculate the cofactor for each element and sum

Cofactor = sign x minor

Signs alternate

Minor = determinant of remaining matrix…

Recursive

Evaluating Determinants 1

BAI

3 1 4

-7 -2 1

9 1 -1

Example: Diagonals

BAI

3 1 4

-7 -2 1

9 1 -1

Example: Cofactors

BAIProperties of Determinants

BAI

• Singular Matrices

Determinant = 0 (otherwise nonsingular)

Row or column of zeros singular

Two rows proportional singular

Properties of Determinants

BAI

• Singular Matrices

Determinant = 0 (otherwise nonsingular)

Row or column of zeros singular

Two rows proprtional singular

Invertible nonsingular

Properties of Determinants

BAI

• Singular Matrices

Determinant = 0 (otherwise nonsingular)

Row or column of zeros singular

Two rows proprtional singular

Invertible nonsingular

• Other properties

Scalar multiple: |cA| = cn|A| (n = matrix dim)

Product: |AB| = |A||B|

Transpose: |At| = |A|

Inverse: |A-1| = 1/|A| (if A-1 exists)

Properties of Determinants

BAI

A and B are 3 x 3 matrices

|A| = -3, |B| = 2

Calculate:

|AB|

|AAt|

|AtB|

|3A2B|

|2AB-1|

|(A2B-1)t|

Example: Properties of Determinants

BAIEvaluating Determinants 2

BAI

• Row Operations and Determinants

1) Multiply by c c|A|

2) Swap two rows -|A|

3) Add multiple of one row to another |A|

Evaluating Determinants 2

BAI

• Row Operations and Determinants

1) Multiply by c c|A|

2) Swap two rows -|A|

3) Add multiple of one row to another |A|

Get zero columns / rows and use cofactors

Evaluating Determinants 2

BAI

• Row Operations and Determinants

1) Multiply by c c|A|

2) Swap two rows -|A|

3) Add multiple of one row to another |A|

Get zero columns / rows and use cofactors

• Numerical Method

Use row ops to get matrix into upper triangular form

Only need 2) and 3)

Keep track of op 2)

Determinant is product of diagonal elements

Zero on diagonal & zeros below singular

Evaluating Determinants 2

BAI

1 0 –2 1

2 1 0 2

-1 1 –2 1

3 1 –1 0

Example: Numerical Evaluation

BAI

1 -1 0 2

-1 1 0 0

2 -2 0 1

3 1 5 -1

Example: Numerical Evaluation

BAIOther Stuff?

BAI

• A-1 = adj(A) / |A|

Adjoint is transpose of matrix of cofactors

Other Stuff?

BAI

• A-1 = adj(A) / |A|

Adjoint is transpose of matrix of cofactors

• System of Equations AX = B

Unique solution A nonsingular

Otherwise, could be many or no solutions

Cramer’s Rule: xi = |Ai| / |A|

Other Stuff?