Sec 3.6
Determinants
Sec 3.6 Determinants
Recall from section 3.5 :
TH2: the invers of 2x2 matrix
1  d  b
A
det  c a 
det  ad  bc
a b 
A

c
d


Sec 3.6 Determinants
2x2 matrix
Example
Evaluate the determinant of
3 5
A

1
2


3 5
 (3)( 2)  (5)(1)  6  5  1
det A 
1 2
How to compute the Higher-order determinants
1 2 1
det B  3 5 1
4 0 2
1
0
det C  0
1
1
0
1
1
0
1
0
1
0
1
1
1
1
1
0
0
0
1
1
0
1
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
1 0 2 
A  4 3 1
3 5 1
M 23 , M 32 , M 33 ,
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
Aij  (1)i  j M ij
Example
Find
1 0 2 
A  4 3 1
3 5 1
A23 , A32 , A33 ,
signs
   
  


   
Sec 3.6 Determinants
3x3 matrix
signs
a11
a12
a21 a22
a31 a32
a13
a23  a11 A11  a12 A12  a13 A13
a33
 a11M 11  a12 M 12  a13M 13
Example
Find det A
1 0 2 
A  4 3 1
3 5 1
   
  


   
Sec 3.6 Determinants
The cofactor expansion of det A along the first
row of A
a11
a12
a21 a22
a31 a32
Note:
 3x3 determinant
 4x4 determinant
 5x5 determinant
 nxn determinant
a13
a23  a11 A11  a12 A12  a13 A13
a33
expressed in terms of
expressed in terms of
expressed in terms of
expressed in terms of
three 2x2 determinants
four 3x3 determinants
five 4x4 determinants
n determinants of size (n-1)x(n-1)
Sec 3.6 Determinants
nxn matrix
det A  a11 A11  a12 A12    a1n A1n
We multiply each element by its cofactor ( in the first row)
Also we can choose any row or column
Th1: the det of an nxn matrix can be obtained
Example
0
2
 0 1
A
7
4

 6 2
0  3
0 0 
3 5

2 4
by expansion along any row or column.
i-th row
det A  ai1 Ai1  ai 2 Ai 2    ain Ain
j-th column
det A  a1 j A1 j  a2 j A2 j    anj Anj
Row and Column Properties
Prop 1: interchanging two rows (or columns)
Example
0
2
 0 1
A
7
4

 6 2
0  3
0 0 
3 5

2 4
 2 3
0
0

B
7
5

 6 4
0 0
0  1
3 4

2 2
det A   det B
Example
0
2
 0 1
A
7
4

 6 2
0  3
0 0 
3 5

2 4
det A   det C
 6 2
 0 1
C
7
4

0
2





0  3
2
0
3
4
0
5
Row and Column Properties
Prop 2: two rows (or columns) are identical
 2 3
0
1

B
7
5

 6 4
Example
Example
0  3
0 1 
3 5

2 4
 6 2
 0 1
C
7
4

 6 2
2 4
0 0
3 5

2 4
det B  0
det C  0
Row and Column Properties
Prop 3:
Example
(k) i-th row + j-th row
0
2
 0 1
A
7
4

 6 2
0  3
0 0 
3 5

2 4
(k) i-th col + j-th col
0
2
 0 1
B
7
4

 6 2
0  3
0  2
3 13 

2 8
det A  det B
Example
0
2
 0 1
A
7
4

 6 2
0  3
0 0 
3 5

2 4
det A  det C
2 0
0  1
C
7 4

2 2
0  3
0 0 
3 5

2  8
Row and Column Properties
Prop 4:
Example
(k) i-th row
0
2
 0 1
A
7
4

 6 2
0  3
0 0 
3 5

2 4
(k) i-th col
0
2
 0 5
B
 7 20

 6 10
0  3
0 0 
3 5

2 4
det B  (5) det A
Example
0
2
 0 1
A
7
4

 6 2
0  3
0 0 
3 5

2 4
det C  (3) det A
0
 2
 0
1
C
 7
4

 18 6
0  3
0 0 
3 5

6 12 
Row and Column Properties
Prop 5:
i-th row B = i-th row A1 + i-th row A2
Example
0
 2
 0
1

B
 7
4

 18 6
0  3
0 0 
3 5

6 12 
0
 2
 0
1

A1 
 7
4

 12 4
0  3
0 0 
3 5

4 10 
0
2
 0 1
A2  
7
4

 6 2
0  3
0 0 
3 5

2 2
det B  det A1  det A2
Prop 5:
i-th col B = i-th col A1 + i-th col A2
Row and Column Properties
upper tria ngular matrix
2 2
0  1
A
0 0

0 0
lower tria ngular matrix
1  3
2 9 
3 5

0 4
Zeros below main diagonal
Prop 6:
Example
2 0
2  1
A
1 3

9 7
0 0
0 0
3 0

4 4
triangular matrix
Either upper
or lower
Zeros above main diagonal
det( triangular ) = product of diagonal
2 2
0  1
A
0 0

0 0
1  3
2 9 
3 5

0 4
Row and Column Properties
Example
2 2
1  1
A
1 6

0 0
1  3
2 9 
3 5

0 4
Transpose
Transpose of a matrix
A  [aij ]
Prop 6:
Example
AT  [a ji ]
1 2 3
A  4 5 6
7 8 9
1 4 7 
AT  2 5 8
3 6 9
det( matrix ) = det( transpose)
1 2 3
A  4 5 6
7 8 9
1 4 7 
B  2 5 8
3 6 9
det A  det B
Transpose
A 
A
cAT
 cAT
T T
A  B
T
 AB T
 A B
T
 B T AT
T
Determinant and invertibility
THM 2:
The nxn matrix A is invertible
1  3
2 9 
0 5

0 4
Example : find A -1
2 2
0  1
A
0 0

0 0
Example : find A -1
2  3 2  3
1  1 1 9 

A
2 6 2 5 


6
4
6
4


detA
=
0
Theorem7:(p193)
A
row equivalent
I
Every n-vector b
Ax = b
det( A)  0
has unique sol
A
Invertible
A
Every n-vector b
is a product of
elementary
matrices
Ax = b
is consistent
The system
Ax = 0
has only the
trivial sol
A
nonsingular
Determinant and inevitability
THM 2:
det ( A B ) = det A * det B
AB  A B
Note:
A
1
1

A
Proof:
Example: compute
1 0
1  1
A
2 6

6 4
0 0
0 0
2 0

6 1
A 1
Cramer’s Rule (solve linear system)
Solve the system
a11x1  a12 x2  a13 x3    a1n xn  b1
(eq 1)
a 21x1  a 22 x2  a 23 x3    a 2n xn  b1
(eq 2)


a n1x1  a n2 x2  a n3 x3    a nn xn  b1
(eq n)
 a1n   x1   b1 
 a2 n   x2  b2 

    
   
 ann   xn  bn 
 b1
b1  a1n
a11
a12
b2 a22  a2 n
   
a21 b2  a2 n
   
a21

a22  b2
  
an 2  ann
A
a
b  ann
x2  n1 n
A
an1
an 2  bn
A
b1
x1 
 a11 a12
a
 21 a22
 

an1 an 2
bn
a12
 a1n
a11

xn 
Sec 3.6 Determinants
Cramer’s Rule (solve linear system)
Solve the system
a11x  a12 y  b1
 a11 a12   x   b1 
 
a



 21 a22   y  b2 
a12
a12
a21 a22
a21x  a22 y  b2
b1
a11
a11
 det A
b1
b2 a22
x
a11 a12
a21 b2
y
a11 a12
a21 a22
a21 a22
Example
Solve the system
3x  5 y  2
x  2y 1
Cramer’s Rule (solve linear system)
Use cramer’s rule to solve the system
x  4 y  5z  1
(eq1)
4 x  2 y  5 z  0 (eq2)
- 3 x  3 y   z  0 (eq3)
Adjoint matrix
Def: Cofactor matrix
Let A =[aij] be an nxn matrix . The cofactor matrix = [Aij]
Example
Find the cofactor matrix
1 0 2 
A  4 3 1
3 5 1
Def: Adjoint matrix of A
Adj A  (cofactor matrix)
Adj A  [Aij ]T  [ Aji ]
T
signs
   
  


   
Example
Find the adjoint matrix
1 0 2 
A  4 3 1
3 5 1
Another method to find the inverse
How to find the inverse of a matrix
Thm2: The inverse of A
Example
Find the inverse of A
1 0 2 
A  4 3 1
3 5 1
A1 
Adj A
A
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
only 40 million operations per second
To evaluate a determinant of a 15x15 matrix using cofactor
15!
expansion  requires
40,000,000
seconds  9.08 Hours
a supercomputer capable of a billion operations per seconds
To evaluate a detrminant of a 25x25 matrix using cofactor expansion  requires
1.55 x1016
1.55x10 25
1.55x10 25
25!
16
sec 
sec  1.55 x10 sec 
sec 
9
9
9
365.25 x 24 x3600
10
10
10
 9.64 x10 47 years