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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
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