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EGR . 1302 - . Determinants. Slide . 2. Determinants. “Eyeball” Method. 3 positive terms. 3 negative terms. - A Property of a Square Matrix. Slide . 3. Determinant of a 3x3. Let’s factor out the elements of the first row of the matrix, i.e.. ID: 756269

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Slide1

Slide 1

Fundamentals of Engineering Analysis

EGR

1302 -

Determinants

Slide2Slide 2

Determinants

“Eyeball” Method

3 positive terms

3 negative terms

- A Property of a Square Matrix

Slide3Slide 3

Determinant of a 3x3

Let’s factor out the elements of the first row of the matrix, i.e.

Slide4Slide 4

Determinant of a 3x3

We can identify this construct as the “Cofactor”

Slide5Slide 5

The Cofactor Matrix of a 3x3

The cofactor of any element is “the determinant formed by striking out the Row & Column of that element

Every element in a square matrix has a

cofactor

Slide6Slide 6

The Cofactor Matrix of a 3x3

Sign of the Cofactor:

Caution: Do not forget the signs of the cofactors

Slide7Slide 7

Determinant by Row Expansion

Row Expansion:

using the first row:

Slide8Slide 8

Using the TI-89 to find Determinants

We had previously entered a matrix

and assigned it to the variable “a”

The calculator has the built-in function “det()“

Which calculates the determinant of a square matrix.

Slide9Slide 9

Determinant by Row or Column Expansion

Select

Any

Row or Column to do the Expansion

Pick Column #1 to simplify the calculation due to the zero terms.

Slide10Slide 10

Finding the Cofactor Matrix of A

Calculators and Computers obviously make this process easier.

Slide11Slide 11

Rules for 2x2 Inverse and the Cofactor Matrix

1. Swap Main Diagonal

2. Change Signs on a

12

, a

213. Divide by detASimilar, but not quite

Slide12Slide 12

Properties of Determinants

1. Determinant of the Transpose Matrix

det A = det A

T

Slide13

Slide 13

Properties of Determinants

2. Multiply a single Row (Column) by a Scalar - k

det B = k*det A

for

det B = 3*det A

Slide14Slide 14

Properties of Determinants

3. If two Rows (Columns) are swapped, the sign changes

det B = -det A

swap

Recall:

4. Expansion by any Rows (Columns) equals the same Determinant

Slide15Slide 15

Properties of Determinants

5. If two Rows (Columns) are equal, or the same ratio,

i.e., Row

1

= k*Row2

det A = 0 det B = 0 Col2 = 2*Col1

det A = 0 Row2 = Row1 The matrix A is “singular”

Recall Rule #3 to find A-1,divide by detA

But if detA=0,a unique solution does not exist

Slide16Slide 16

Properties of Determinants

If a new matrix B is constructed from A

by adding K*row

j

to another rowi …

det B = det A

Construct D by creating a new Row 2

These are called Row (Column) Operations

Slide17Slide 17

Finding the Determinant: Two Methods

2 + (-40) + (-6) – (-5) -12 –(-8) = -43

“Eyeball” Method

Row Expansion

1*(2-12) -2(-4+3) -5(8-1)

-10 + 2 -35 = -43

Slide18Slide 18

Questions?

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