Sec 3.6 Determinants

Example Evaluate the determinant of

21

53A

21

53det A )1)(5()2)(3( 156

2x2 matrix

Sec 3.6 Determinants

Example Solve the system

12

253

yx

yx1

21

53det A

Cramer’s Rule (solve linear system)

1

2

21

53

y

x

Sec 3.6 Determinants

Solve the system

22221

11211

byaxa

byaxa

Cramer’s Rule (solve linear system)

2

1

2221

1211

b

b

y

x

aa

aa

Aaa

aadet

2221

1211

A

ab

ab

xdet

222

121

A

ba

ba

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

153

134

201

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

153

134

201

A

,,, 333223 AAA

ijji

ij MA )1(

signs

Sec 3.6 Determinants

131312121111

333231

232221

131211

AaAaAa

aaa

aaa

aaa

3x3 matrix

131312121111 MaMaMa

signs

Example Find det A

153

134

201

A

Sec 3.6 Determinants

131312121111

333231

232221

131211

AaAaAa

aaa

aaa

aaa

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)

4226

5347

0010

3002

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

4226

5347

0010

3002

A

2246

4357

1000

0032

B

BA detdet

Example

4226

5347

0010

3002

A

CA detdet

3002

5347

0010

4226

C

Row and Column Properties

Prop 2: two rows (or columns) are identical

Example

4246

5357

1010

3032

B 0det B

Example

0det C

4226

5347

0010

4226

C

Row and Column Properties

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

Example

4226

5347

0010

3002

A

8226

13347

2010

3002

B

BA detdet

Example

4226

5347

0010

3002

A

CA detdet

8222

5347

0010

3002

C

Row and Column Properties

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

Example

4226

5347

0010

3002

A

AB det)5(det

Example

4226

5347

0010

3002

A

AC det)3(det

42106

53207

0050

3002

B

126618

5347

0010

3002

C

Row and Column Properties

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

Example

21 detdetdet AAB

2226

5347

0010

3002

2A

126618

5347

0010

3002

B

104412

5347

0010

3002

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

4000

5300

9210

3122

A

Zeros below main diagonal

matrixngular lower tria

4479

0331

0012

0002

A

Zeros above main diagonal

matrix triangular

Either upper or lower

Example

4000

5300

9210

3122

A

Row and Column Properties

Example

4000

5361

9211

3122

A

Transpose

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

matrix a of Transpose

987

654

321

A

Example

963

852

741TA][ ijaA ][ ji

T aA

987

654

321

A

963

852

741

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

4000

5000

9210

3122

A

-1A find :Example

4646

5262

9111

3232

A

Determinant and inevitability

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

BAAB

Note:

AA

11 Proof:

Example: compute 1A

1646

0262

0011

0001

A

Solve the system

Cramer’s Rule (solve linear system)

n) (eq aa aa

2) (eq aa aa

1) (eq aa aa

1nn3n32n21n1

12n323222121

11n313212111

bxxxx

bxxxx

bxxxx

n

n

n

nnnnnn

n

n

b

b

b

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

A

aab

aab

aab

x nnnn

n

n

2

2222

1121

1 A

aba

aba

aba

x nnnn

n

n

1

2221

1111

2 A

baa

baa

baa

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

zyx

zyx

zyx

Adjoint matrix

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

Example Find the cofactor matrix

153

134

201

A

signs

Def: Adjoint matrix of A Tmatrix)(cofactor AAdj

][][A Tij jiAAAdj

Example Find the adjoint matrix

153

134

201

A

Another method to find the inverse

Thm2: The inverse of A

Example Find the inverse of A

153

134

201

A

A

AAdjA

1

Computational Efficiency

The 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 47

1616

9

25

91064.9

36002425.365

1055.1sec1055.1sec

10

1.55x10 sec

10

!25

Monday

Quiz