Sec 3.6 Determinants

Example Evaluate the determinant of

2153

A

2153

det A )1)(5()2)(3( 156

2x2 matrix

Sec 3.6 Determinants

Example Solve the system

12253

yxyx

12153

det A

Cramer’s Rule (solve linear system)

12

2153yx

Sec 3.6 Determinants

Solve the system

22221

11211

byaxabyaxa

Cramer’s Rule (solve linear system)

2

1

2221

1211

bb

yx

aaaa

Aaaaa

det2221

1211

Aabab

xdet

222

121

Ababa

ydet

221

111

Sec 3.6 Determinants

Def: Minors Let A =[aij] be an nxn matrix . The ijth minor of A ( or the minor of aij) is the determinant Mij of the (n-1)x(n-1) submatrix after you delete the ith row and the jth column of A.

Example Find

153134201

A

,,, 333223 MMM

Sec 3.6 Determinants

Def: Cofactors Let A =[aij] be an nxn matrix . The ijth cofactor of A ( or the cofactor of aij) is defined to be

Example Find

153134201

A

,,, 333223 AAA

ijji

ij MA )1(

signs

Sec 3.6 Determinants

131312121111

333231

232221

131211

AaAaAaaaaaaaaaa

3x3 matrix

131312121111 MaMaMa

signs

Example Find det A

153134201

A

Sec 3.6 Determinants

131312121111

333231

232221

131211

AaAaAaaaaaaaaaa

The cofactor expansion of det A along the first row of A

Note: 3x3 determinant expressed in terms of three 2x2 determinants 4x4 determinant expressed in terms of four 3x3 determinants 5x5 determinant expressed in terms of five 4x4 determinants nxn determinant expressed in terms of n determinants of size (n-1)x(n-1)

Sec 3.6 Determinants

nnAaAaAaA 1112121111det

nxn matrix

Example

We multiply each element by its cofactor ( in the first row)

4226534700103002

A

Also we can choose any row or column

Th1: the det of an nxn matrix can be obtained by expansion along any row or column.

ininiiii AaAaAaA 2211det

njnjjjjj AaAaAaA 2211det

i-th row

j-th row

Row and Column Properties

Prop 1: interchanging two rows (or columns)

Example

4226534700103002

A

224643571000

0032

B

BA detdet

Example

4226534700103002

A

CA detdet

3002534700104226

C

Row and Column Properties

Prop 2: two rows (or columns) are identical

Example

4246535710103032

B 0det B

Example

0det C

4226534700104226

C

Row and Column Properties

Prop 3: (k) i-th row + j-th row (k) i-th col + j-th col

Example

4226534700103002

A

822613347

20103002

B

BA detdet

Example

4226534700103002

A

CA detdet

8222534700103002

C

Row and Column Properties

Prop 4: (k) i-th row (k) i-th col

Example

4226534700103002

A

AB det)5(det

Example

4226534700103002

A

AC det)3(det

421065320700503002

B

126618534700103002

C

Row and Column Properties

Prop 5: i-th row B = i-th row A1 + i-th row A2

Example

21 detdetdet AAB

2226534700103002

2A

126618534700103002

B

104412534700103002

1A

Prop 5: i-th col B = i-th col A1 + i-th col A2

Row and Column Properties

Prop 6: det( triangular ) = product of diagonal

matrixngular upper tria

4000530092103122

A

Zeros below main diagonal

matrixngular lower tria

4479033100120002

A

Zeros above main diagonal

matrix triangular

Either upper or lower

Example

4000530092103122

A

Row and Column Properties

Example

4000536192113122

A

Transpose

Prop 6: det( matrix ) = det( transpose)

matrix a of Transpose

987654321

A

Example

963852741

TA][ ijaA ][ jiT aA

987654321

A

963852741

B BA detdet

Transpose

AATT

TTT BABA

TT cAcA

TTT ABAB

Determinant and invertibility

THM 2: The nxn matrix A is invertible detA = 0

-1A find :Example

4000500092103122

A

-1A find :Example

4646526291113232

A

Determinant and inevitability

THM 2: det ( A B ) = det A * det B

BAAB

Note:

AA 11 Proof:

Example: compute 1A

1646026200110001

A

Solve the system

Cramer’s Rule (solve linear system)

n) (eq aa aa

2) (eq aa aa1) (eq aa aa

1nn3n32n21n1

12n323222121

11n313212111

bxxxx

bxxxxbxxxx

n

n

n

nnnnnn

n

n

b

bb

x

xx

aaa

aaaaaa

2

1

2

1

21

22221

11211

Aaab

aabaab

x nnnn

n

n

2

2222

1121

1 Aaba

abaaba

x nnnn

n

n

1

2221

1111

2 Abaa

baabaa

x nnnn

21

22221

11211

Use cramer’s rule to solve the system

Cramer’s Rule (solve linear system)

(eq3) 033-(eq2) 0524

(eq1) 15 4

zyxzyxzyx

Adjoint matrixDef: Cofactor matrix

Let A =[aij] be an nxn matrix . The cofactor matrix = [Aij]

Example Find the cofactor matrix

153134201

A

signs

Def: Adjoint matrix of A Tmatrix)(cofactor AAdj

][][A Tij jiAAAdj

Example Find the adjoint matrix

153134201

A

Another method to find the inverseThm2: The inverse of A

Example Find the inverse of A

153134201

A

AAAdjA 1

Computational EfficiencyThe amount of labor required to compute a numerical calculation is measured by the number of arithmetical operations it involves

Goal: let us count just the number of multiplications required to evaluate an nxn determinant using cofactor expansion

2x2: 2 multiplications

3x3: three 2x2 determinants 3x2= 6 multiplications

4x4: four 3x3 determinants 4x3x2= 24 multiplications

5x5: four 3x3 determinants 4x3x2= 24 multiplications

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

nxn: n (n-1)x(n-1) determinants nx…x3x2= n! multiplications

Computational Efficiency

Goal: let us count just the number of multiplications required to evaluate an nxn determinant using cofactor expansion

nxn: determinants requires n! multiplications

a typical 1998 desktop computer , using MATLAB and performing aonly 40 million operations per second

To evaluate a determinant of a 15x15 matrix using cofactor expansion requires Hours 9.08 seconds

000,000,40!15

a supercomputer capable of a billion operations per seconds

To evaluate a detrminant of a 25x25 matrix using cofactor expansion requires

yearsxxx

xx 4716

169

25

9 1064.936002425.365

1055.1sec1055.1sec10

1.55x10 sec 10

!25

Monday

Quiz