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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
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