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DETERMINANTS


2/26

THE LAPLACE EXPANSION

When we have a matrix with dimension larger than 2, we use theLaplace expansionin order to calculate its determinant. For that

we need to introduce two new concepts:


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The minor determinant

The minor determinant thatcorresponds to an element is given bydeleting the row and column of the

element.


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element.

2 311

1 4

5 1 2

3 2 3

8 1 4

The minor

determinantcorresponding tothe 5 is 11.


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element.

1 22

1 4

The minor

determinantcorresponding tothe -3 is 2.

5 1 2

3 2 3

8 1 4


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7/26

Finding Inverses 3x3

An algorithm can be followed to find theinverse of a 3x3 matrix, M.

1. Find the matrix of minor determinants.2. Alter the signs of the minors which

dont lie on the diagonals.

3. Transpose

4. Divide by det(M)


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9/26

LAPLACE EXPANSION METHOD

To find the first row minors of A we work along row 1:

2)7(1)3(331

73

11

M

2)7(2)3(4

32

7412

M

2)3(2)1(412

3413

M

Example

312

734

142

A


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LAPLACE EXPANSION METHOD

Cofactors: The cofactor Cijis equal to the corresponding minor, Mijwith a prescribed algebraic sign attached to it. If the sum of i (the

number indicating the row) and j(the number indicating the column

is even (e.g., 2,4,8) then the cofactor has the same sign as the

minor. If the sum is odd(e.g, 3, 5) then the cofactor has the

opposite sign:

333231

232221

131211

333231

232221

131211

MMM

MMM

MMM

CCC

CCC

CCC


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LAPLACE EXPANSION METHOD To calculate the determinant we simply multiply each element of

the chosen row (or column) with its corresponding cofactor andsum the results:

131312121111131312121111

333231

232221

131211

MaMaMaCaCaCaA

aaa

aaa

aaa

A


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THE INVERSE OF A MATRIX OF HIGHER

DIMENSION

The Laplace expansion introduced us to the concept ofminors and cofactors. We use these concepts in the

calculation of the inverse of a matrix.

Specifically,

333231

232221

131211

333231

232221

131211

333231

232221

131211

MMM

MMM

MMM

CCC

CCC

CCC

C

aaa

aaa

aaa

A


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THE INVERSE OF A MATRIX OF HIGHER

DIMENSION

The transpose of the cofactor matrix is called the adjoint matrix:

adjA=C

General case of inverse matrix calculation: Provided that the

determinant of the matrix is non-zero, the inverse of any (nxn) matrix

A can be found by multiplying its adjoint matrix by 1 divided by the

determinant.

Example: Find the inverse of

703

230

114

A


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Solution

The inverse can be found as follows:

,

703

230

114

A ,2102170

2311 C 6)60(

73

2012 C

,99003

3013 C

7)07(70

1121

C 3132873

1422

C

,3)30(03

1423 C ,532

23

1131

C

8)08(20

1432 C ,12

30

1433

C,

1285

3317

9621

Cor

TCadjA1239

8316

5721

131312121111 CaCaCaA 099)9(1)6(1)21(4

1239

8316

5721

99

11

AHence


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More examples of matrix inversion

231

312

541

,

121

231

112

BA

Find where possible the inverse of:


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The Cramers rule

Now that we know how to find the inverse of a matrix, we

can solve a related system of equations in exactly the

same way as in the 2X2 case, x=A-1b. But as we saw this

is a very time and effort consuming process, involving the

calculation and subsequent inversion of the co-factor

matrix and evaluation of the determinant.


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18/26

The Cramers rule cntnd.

According to the Cramers rule, the solution for (x1, x2and x3) is

given by:

33231

22221

11211

3

33331

23221

13111

2

33323

23222

13121

1

3

3

2

2

1

1

,,

,,,

baa

baa

baa

A

aba

aba

aba

A

aab

aab

aab

A

whereA

Ax

A

Ax

A

Ax


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The Cramers rule

Example: Solve by Cramers rule:

Step 1:

13572

964

932

321

321

321

xxx

xxx

xxx

,

572

614

321

A ,

3

2

1

xx

x

x

139

9

b

,

5713

619

329

1 A

,

5132

694

391

2 A

1372

914

921

3 A


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Cramers Rule - 3 x 3

Consider the 3-equation system below with

variables x, yand z:

a1xb

1yc

1zC

1

a2xb2y

c2z

C2

a3xb

3yc

3zC

3


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THE CRAMERS RULE

Step 2: Calculate321 ,,, AAAA

6372

143

52

642

57

611

572

614

321

A

315713

193513

69257

619

5713

619

329

1 A

63132

943

52

649

513

691

5132

694

391

2

A

12672

149

132

942

137

911

1372

914

921

3

A


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THE CRAMERS RULE

Step 3: Calculate x1, x2and x3

263

126,1

63

63,5

63

315 33

2

2

1

1

A

Ax

A

Ax

A

Ax


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Cramers Rule - 3 x 3

The formulae for the values of x, yand zare shownbelow. Notice that all three have the samedenominator.

x

C1 b

1 c

1

C2 b

2 c

2

C3 b

3 c

3

a1 b

1 c

1

a2 b

2 c

2

a3 b

3 c

3

y

a1 C

1 c

1

a2 C

2 c

2

a3 C

3 c

3

a1 b

1 c

1

a2 b

2 c

2

a3 b

3 c

3

z

a1 b

1 C

1

a2 b

2 C

2

a3 b

3 C

3

a1 b

1 c

1

a2 b

2 c

2

a3 b

3 c

3


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Cramers Rule

Not all systems have a definite solution. If thedeterminant of the coefficient matrix is zero, a solutioncannot be found using Cramers Rule

due to division by zero.


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Cramers Rule

Example:Solve the system 3x- 2y+ z= 9

x+ 2y- 2z= -5x+ y - 4z= -2

x

9 2 1

5 2 2

2 1 43 2 1

1 2 2

1 1 4

2323

1 y

3 9 1

1 5 2

1 2 43 2 1

1 2 2

1 1 4

6923

3


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Cramers Rule

Example, continued: 3x- 2y+ z= 9

x+ 2y- 2z= -5

x+ y- 4z= -2

z

3 2 9

1 2 5

1 1 2

3 2 11 2 2

1 1 4

0

23 0

The solution is

(1, -3, 0)

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