17 Determinants
2019/04/25
17.1 Concepts
Many problems in physical sciences, in engineering , and in statistics give rise to
system of simultaneous linear equations. In some cases however the number of
equations can be large, and alternative methods are then required both for the
numerical solution of the large “linear system”, and for the formulation and
theoretical analysis of the problems that give rise to them.
Some of the practical methods of solution, the “numerical methods”, are discussed in
Chapter 20. The branch of mathematics concerned with the theory of linear systems
Is matrix algebra, the subject of Chapters 18 and 19, but several of the more
important and useful results in the theory of linear equations can be derived
independently from the quantities called determinants.
The theory of determinants is discussed in this chapter as a separate topic, partly
because determinants have certain symmetry properties that have made them an
important tool in quantum mechanics. They are used in quantum chemistry to
construct electronic wave functions that are consistent with the requirements of the
Pauli Exclusion Principle.
Then concept of determinants has its origin in the solution of simultaneous linear equations.
(1) a1 x  b1 y  c1
Linear equations
(17.1)
(2) a2 x  b2 y  c2
Sol. I (1) a1b2 x  b1b2 y  c1b2
(2) b1a2 x  b1b2 y  b1c2
(1)  (2) : (a1b2  b1a2 ) x  c1b2  b1c2 ,
c1b2  b1c2
x 
;
a1b2  b1a2
Sol. II
D
a1
b1
a2
b2
a1
b1
a2
b2
a1c2  c1a2
y
a1b2  b1a2
 a1b2  b1a2
, D1 
c1
b1
c2
b2
D1
x
,
D
, D2 
a1
c1
a2
c2
D2
y
D
Example 17.1
The determinant is a property of a square array of 4=22 elements, the coefficients of x
and y in the system of equations (17.1). It is a determinant of order 2. In the general
case, a determinant of order n is a property of a square array of n2 elements, and is
written
column
Element:
aij
The ith row and jth column of the array
a11
a12
 a1n
a21
a22
 a2 n


an1
row

an 2  ann
Determinants of order n arise from the consideration of system of order n linear
equations in n unknowns.
17.2 Determinants of order 3
The system of three linear equations in three unknowns, x1, x2, and x3.
(1) a11 x1  a12 x2  a13 x3  b1
(2) a21 x1  a22 x2  a23 x3  b2
(3) a31 x1  a32 x2  a33 x3  b3
a11
a12
a13
D  a21
a22
a23  a11
a31
a32
a33
D1
D2
x1 
, x2 
,
D
D
b1 a12 a13
a22
a23
a32
a33
D3
x3 
D
a11 b1
 a21
a12
a13
a32
a33
 a31
a12
a13
a22
a23
Example 17.2
a13
a11
a12
b1
D1  b2
a22
a23 , D2 a21 b2
a23 , D3  a21
a22
b2
b3
a32
a33
a33
a32
b3
a31
b3
a31
Order n-1
Minors and cofactors
Mij
a11
a12
a13
a21
a22
a23 
a31
a32
a33
a11
a12
a31
a32
 M 23
They are important because they used for the expansion of a determinant in terms of
its elements.
a11
a12
a13
a21
a22
a23  a11M 11  a21M 21  a31M 31
a31
a32
a33
This is called expansion along the first column.
The sign associated with element aij is
  1 if i  j is even
i j
(1)  
 1 if i  j is odd
  
The signs for the first-order
determinant are
  
  
Example 17.3
Example 17.4
A determinant can be expanded along
any row or column.
Cofactors
Cij
Cij  (1) i  j M ij
The expansion along with the first column is then
a11
a12
a13
a21
a22
a23  a11C11  a21C21  a31C31   ai1Ci1
a31
a32
a33
3
i 1
More generally, the expansion along with the jth column is
a11
a12
a13
a21
a22
a23  a1 j C1 j  a2 j C2 j  a3 j C3 j   aij Cij
a31
a32
a33
and along the ith row is
a13
3
i 1
An expansion in terms of cofactors is called Laplace
expansion of the determinant.
a11
a12
a21
a22
a23  ai1Ci1  ai1Ci1  ai1Ci1   aij Cij
a31
a32
a33
3
j 1
Example 17.5 Find the cofactors of the elements of the third column of
2
4
1
1
3
2 0
1
Cij  (1) i  j M ij
0
We have
a13  3 M 13 
a23  0 M 23 
4
2
1
1
2
1
1 1
 2, C13  (1)13 M 13  2
 3, C23  (1) 23 M 23  3
17.3 The general case
A determinant of order n is a property of a square array of n2 elements, written
For n=1:
D  a11  a11
 a1n
a11
a12
a13
a21
a22
a23  a2 n
D  a31
a32
a33  a3n


an1
an 2


an 3  ann
For n  2, the expansion along any row i (i= 1, 2, …, n):
n
D  ai1Ci1  ai 2Ci 2    ain Cin   aij Cij
j 1
The expansion along any row j (j= 1, 2, …, n):
n
Example 17.6
D  a1 j C1 j  a2 j C2 j    anj Cnj   aij Cij , Cij  (1) i  j M ij
i 1
Example 17.7 Find the value of the “triangular” determinant
1 2
3
4
5
0 6
7
8
9
D  0 0 10 11 12  1
0 0
0
13 14
0 0
0
0
15
6
7
8
9
0 10 11 12
0
0
13 14
0
0
0
15
10 11 12
 1 6 0
0
13 14
0
15
 1 6 10 13 15
The value of a triangular determinant is equal to the product of the elements on the
diagonal,
D  a11  a22  a33    ann
Note: on the valuation of determinants and the solution of linear equations.
The expansion of a determinant in terms of its elements consists of n! products of n
elements at a time, and involves n!(n-1) multiplications. It follows therefore that such
expansions do not provide a practical (or accurate) method for the evaluation of large
determinants. The same is true of the use of Cramer’s rule for the solution of linear
equations.
17.4 The solution of linear equations
In general, the system of n linear equations,
a11 x1  a12 x2  a13 x3    a1n xn  b1
a21 x1  a22 x2  a23 x3    a2 n xn  b2
a31 x1  a32 x2  a33 x3    a3n xn  b3





an1 x1  an 2 x2  an 3 x3    ann xn  bn
a11
a12
a13
(17.26)
This solution is given by Cramer’s rule.
Dn
D1
D2
x1 
, x2 
,  , xn 
D
D
D
 a1n
a11
a21
a22
a23  a2 n
If D0
D  a31
a32
a33  a3n
x2 
(17.27)


an1
an 2


an 3  ann
a13
 a1n
a21 b2
a23  a2 n
b3
a33  a3n
D2
 a31
D

(17.28)
b1

an1 bn


an 3  ann
Cramer’s rule applies even when all the quantities bk on the right sides of equations
(17.26) are zero. In this case, all the determinants Dk are also zero and the solution is
x1  x2  x3    xn  0
The case D=0
Cramer’s rule provides the unique solution of the system of linear equations (17.26) so
long as the determinant of the coefficient (17.27) is not zero. The rule does not apply
however when D=0 because division by D=0 in (17.28) is not allowed. There is then in
general no unique solution or, in some case, no solution of the equations at all.
(1) 2 x  2 y  z  10
(2)
x  2 y  2 z  3
(3) 3 x  2 y  4 z  20
2 2
1
D  1 2 2  0
3 2
4
(2)  (3) : 4 x  4 y  2 z  17
(1)  2 : 4 x  4 y  2 z  20
The case D=0
No unique solution
(no solution of the equation at all)
(in consistent)
(1) 2 x  2 y  z  10
( 2)
x  2 y  2 z  3
(linearly dependent = linear combination of each other)
(3) 3 x  2 y  4 z  23
(1)  (2) : x  13  2 z ,
This is a solution of the system for every value of z.
1
(2)  2  (1) : y  (5 z  16)
2
Homogeneous equations
a11 x1  a12 x2  a13 x3    a1n xn  0
a21 x1  a22 x2  a23 x3    a2 n xn  0
a31 x1  a32 x2  a33 x3    a3n xn  0




Homogeneous  inhomogeneous eq.
If D0, trivial (zero) solution x=y=z=0

an1 x1  an 2 x2  an 3 x3    ann xn  0
Only the zero solution exists if D0, but other solutions also exist when D=0.
(1) 2 x  2 y  z  0
( 2)
x  2 y  2z  0
(3) 3 x  2 y  4 z  0
(1)  (2) : x  3 z ,
5z
(2)  2  (1) : y 
2
has D=0, the equations are linearly dependent.
Nonzero solutions are obtained by solving any pair of
the equations for x and y in terms of z:
One of the most important theorem of systems of linear
equations:
A system of homogeneous linear equations has non trivial
solution only if the determinant of the coefficients is zero.
Secular equations
A number of problems in the physical sciences give rise to systems of equations of the form
a11 x1  a12 x2  a13 x3    a1n xn  x1
 Is a parameter to be determined.
a21 x1  a22 x2  a23 x3    a2 n xn  x2
a31 x1  a32 x2  a33 x3    a3n xn  x3





an1 x1  an 2 x2  an 3 x3    ann xn  xn
In molecular-orbital theory, the Schrödinger equation is replaced by such a set of linear
equations in which the quantities x1, x2, … , xn represents an orbital and  the
corresponding orbital energy.
(a11   ) x1  a12 x2  a13 x3    a1n xn  0
a21 x1  (a22   ) x2  a23 x3    a2 n xn  0
a31 x1  a32 x2  (a33   ) x3    a3n xn  0





an1 x1  an 2 x2  an 3 x3    (ann   ) xn  0
These homogeneous equations, called secular equations, have non-trivial solution only
of the determinant of the coefficients is zero.
(a11   )
a12
a13

a1n
a21
(a22   )
a23

a2 n
a31
a32
(a33   ) 
a3n



an1
an 2
an 3
0
(17.39)

 (a11   )
The determinant is called a secular determinant in this context. It is zero only for
some values of the parameter  and these are obtained by solving equation (17.39).
Because the expansion of the secular determinant is a polynomial of degree n in ,
the required values of  are the n roots of the polynomial.
Example 17.8 Find the values of  for which the following system of equations has nonzero
solution:
 2 x  y  z  x
 11x  4 y  5 z  y
 z
x y
(2   ) x  y  z  0
 11x  (4   ) y  5 z  0
 x  y  (  ) z  0
2
D   11
1
1
4
1
1
5  3  22    2  (  1)(  1)(  2)

 D  0,   1,  1, and 2.
Because of the n roots of the secular determinant three exists a solution of the secular
equations.
Example 17.9 Solve the Hückel molecular-orbital problem for the allyl radical CH2CHCH2
in terms of the Hückel parameters  and :
(1) (  E )c1   c2
0
(2)  c1  (  E )c2  c3  0
c2  (  E )c3  0
(3)
 E

0

 E
  (  E )(  E ) 2  2  2   0
0

 E
E1   , E2    2  and E3    2 
For
E  E1   : (1) c2  0, c2  0, (3) c2  0, c2  0
(2) c1  c3  0, c1  c3
Setting c3=1:
E
c1
c2
c3

1
0
1
  2
  2
1
1
2
 2
1
1
化二乙
17.5 Properties of determinants
The following are the more important general properties of determinants.
1. Transposition
The value of a determinant is unchanged if its rows and columns are interchanged:
a1
b1
c1
a1
a2
a3
a2
b2
c2  b1
b2
b3
a3
b3
c3
c2
c3
c1
Column 1  row 1
Column 2  row 2
Column 3  row 3
2. Multiplication by a constant
If all the elements of any row (or column) are multiplied by the same factor , the
value of the new determinant is  times the value of the old determinant:
a1 b1 c1
a1
a2
a3
a2
b2
c2   b1
b2
b3
a3
b3
c3
c2
c3
c1
If   0,
0
0
0
a2
b2
c2  0
a3
b3
c3
Example 17.10
3. Addition rule
If all the elements of any row (or column) are written as the sum of two terms the determinant
can be written as the sum of two determinants:
a1  d1
b1
c1
a1
b1
c1
d1
b1
c1
a2  d 2
b2
c2  a 2
b2
c2  d 2
b2
c2
a3  d 3
b3
c3
b3
c3
b3
c3
a3
d3
4. Interchange of rows (or columns). Antisymmetry
If two rows (or columns) of a determinant are interchanged the value of the determinant is
multiplied by (-1):
a b c
a b c Interchange of rows 1 and 2
1
1
a2
1
2
2
2
b2
c2   a1
b1
c1
a3
b3
c3
a3
b3
c3
a1
b1
c1
a1
c1
b1
Interchange of columns 2 and 3
a2
b2
c2   a 2
c2
b2
a3
b3
c3
c3
b3
It is this property of determinants that has made
them so useful for the construction of electronic
wave functions.
a3
Example 17.11
Example 17.12
5. Two rows or columns equal (related to property 4)
The value of a determinant is zero if two rows (or two columns) are equal:
a1
b1
c1
a1
b1
c1  0
a3
b3
c3
If the rows (or columns) are identical then such an
interchange must leave the determinant unchanged.
Therefore –D=D and this is possible only if D=0
Example 17.13
6. Proportional rows or columns (related to property 2+5)
The value of a determinant is zero if one row (or column) is a multiple of another row (or
column):
Example 17.14
a b c
a b c
2
2
2
2
2
2
a2
b2
c2   a 2
b2
c2
a3
b3
c3
b3
c3
a3
Example 17.15
7. Addition of rows or columns (related to property 3+6)
The value of a determinant is unchanged when a multiple of any row ( or column) is
added to any other row (or column):
a1  b1
b1
c1
a1
b1
a2  b2
b2
c2  a 2
b2
a3  b3
b3
c3
b3
a3
b1 b1 c1 a1 b1 c1
c2  b2 b2 c2  a2 b2 c2
c3 b3 b3 c3 a3 b3 c3
c1
8. Linearly-dependent rows or columns (related to property 5+6+7)
The value of a determinant is zero if the rows (or columns) are linearly dependent; that is, if
a row (or column) is a linear combination of the others:
b1  c1 b1 c1
b1 b1 c1
c1 b1 c1
b2  c2 b2 c2   b2 b2 c2   c2 b2 c2  0
b3  c3 b3 c3
b3 b3 c3
c3 b3 c3
9. Derivative of a determinant
If the elements of a determinant D are differentiate functions, the derivative D’ of D can be
written:
D  D1  D2  D3    Dn
where Di is obtained from D by differentiation of the elements of the ith row.
D  D1  D2  D3
Example 17.16
a1
d
b1
dx
c1
b2
da1
a3
dx
b3  b1
da2
dx
b2
c2
c3
c2
a2
c1
da3
a1
dx
db1
b3 
dx
c3
c1
a2
db2
dx
c2
a3
a1
db3
 b1
dx
dc1
c3
dx
a2
a3
b2
dc2
dx
b3
dc3
dx
17.6 Reduction to triangular form
A “triangular” determinant has value equal to the product of its diagonal elements,
a11
a12
a13  a1n
0
a22
a23  a2 n
0
0



0
0
0
a33  a3n  a11  a22  a33    ann

 ann
Every determinant can be reduced to triangular form
by means of a systematic application of property 7 in
Section 17.5. The method is an example of elimination
methods discussed in Chapter 20.
Example 17.17
1 2 3
1
2
3
1
2
3
1
2
3
3 2 5  0  4  4  0  4  4  0  4  4  1 (4) 1  4
2 3 6
-3row 1
2
3
-2row 1
6
0
1
0
-1/4row 2
0
0
1
17.7 Alternating functions
A function f ( x1 , x2 , x3 ,  , xn ) of n variables is called an alternating function, or
totally antisymmetric, if the interchange of any two of the variables has the effect of
multiplying the value of the function by (-1).
f ( x1 , x2 , x3 ,  , xn )   f ( x2 , x1 , x3 ,  , xn )
If two variables are equal the function is zero,
f ( x1 , x1 , x3 ,  , xn )  0
A determinant is an alternating function of its rows (columns).
f1 ( x1 )
f1 ( x2 )
f1 ( x3 ) 
f1 ( xn )
f 2 ( x1 )
f 2 ( x2 )
f 2 ( x3 ) 
f 2 ( xn )
f 3 ( x1 )
f 3 ( x2 )
f 3 ( x3 ) 
f 3 ( xn )


f n ( x1 )
f n ( x2 )

f n ( x3 ) 
An alternating function of n variables
has the form with f1 , f 2 ,  , f n
(property 4, P. 490)

f n ( xn )
The interchange of any pair of variables leads to the interchange of two columns and,
therefore, to a change of sign.
For n=2,
f1 ( x1 )
f1 ( x2 )
f 2 ( x1 )
f 2 ( x2 )
2! 2
 f1 ( x1 ) f 2 ( x2 )  f1 ( x2 ) f 2 ( x1 )
3! 6
For n=3,
f1 ( x1 )
f1 ( x2 )
f 2 ( x1 )
f 2 ( x2 )
f 3 ( x1 )
f 3 ( x2 )
 f1 ( x1 ) f 2 ( x2 ) f 3 ( x3 )  f1 ( x1 ) f 2 ( x3 ) f 3 ( x2 )

f 2 ( x3 )   f1 ( x2 ) f 2 ( x3 ) f 3 ( x1 )  f1 ( x2 ) f 2 ( x1 ) f 3 ( x3 )
f 3 ( x3 )  f1 ( x3 ) f 2 ( x1 ) f 3 ( x2 )  f1 ( x3 ) f 2 ( x2 ) f 3 ( x1 )
f1 ( x3 )
The expansion of the determinant has n! products of the functions
The 3!  6 permutations of x1, x2, and x3 are
f1 , f 2 ,  , f n
x1 x2 x3 , x1 x3 x2 , x2 x3 x1 , x2 x1 x3 , x3 x1 x2 , x3 x2 x1
Each term contributes to the sum with + sign if the permutation is obtained from x1x2x3 by
an even number of transpositions, and with – sign for an odd number of transpositions.
Alternating functions in the form of single determinants or sums of determinants are
important in quantum mechanics for the construction of electronic wave functions.
The electron is a member of the class of particles called fermions, particles with halfintegral spin (a particle with zero or integral spin is called boson). The wave function
of a system of identical fermions is totally antisymmetric with respect to the
interchange of the coordinates (including spin) of the fermions. The interchange of the
coordinates of any pair of fermions results in the change of sign of the wave function.