Matrices: the determinant
For example:
0
1
-14.0000
B
A = @ 4.0000
13.0000
Introduction
25.0000
+ (¡12) ¢ 4 ¢ 25 ¡ (¡12) ¢ (¡1) ¢ (13)
Given a 2 £ 2 matrix:
A=
= 154 + 1400 + 364 ¡ 308 ¡ 1200 ¡ 156 = 254
Ã
a11
a21
a12
a22
2.1
!
(1)
the determinant is evaluated as follows:
for example:
-11.0000
-20.0000
19.0000
5.0000
!
jAj = ¡11 ¢ 5 ¡ 19 ¢ (¡20)
= ¡55 + 380 = 325
Given a 3 £ 3 matrix:
a11
B
A = @ a21
a31
a12
a22
a32
1
a13
C
a23 A
a33
jAj = a11 a22 a33 ¡ a11 a23 a32 + a12 a23 a31 ¡ a12 a21 a33
+a13 a21 a32 ¡ a13 a22 a31
Richard Ti¢n
(11)
(12)
Every element in a matrix has a minor. It is found by computing the determinant
of the matrix which remains when the row and column in which the element
in question lies are deleted. For the matrix in 6 the minor of a21 is found by
deleting the second row and the …rst column of the matrix:
M21 =
Ã
(10)
Minors
(2)
det A = jAj = a11 a22 ¡ a12 a21
A =
(9)
Obviously the problem of evaluating a determinant becomes much more
complex as the dimension of the matrix increases. In order to simplify matters
we introduce the concepts of minors and co-factors.
The determinant
0
1
-12.0000
C
4.0000 A
11.0000
jAj = ¡14 ¢ (¡1) ¢ 11 ¡ (¡14) ¢ 4 ¢ 25 + 7 ¢ 4 ¢ 13 ¡ 7 ¢ 4 ¢ 11
The determinant is a single number that is associated with a matrix of whatever
dimension and the inverse is computed from a matrix using the determinant.
We will examine the inverse in a later lecture.
2
7.0000
-1.0000
January 30, 2001
(3)
a12
a32
a13
a33
!
jM21 j = a12 a33 ¡ a13 a32
(15)
For example the minors of:
0
-4.0000
B
A = @ -12.0000
-1.0000
(7)
(8)
1
(14)
Similarly the minor of a23 is:
jM23 j = a12 a32 ¡ a12 a31
(6)
(13)
and computing the determinant as follows:
(4)
(5)
Ã
Richard Ti¢n
0.0000
4.0000
16.0000
January 30, 2001
1
-20.0000
C
2.0000 A
1.0000
(16)
2
are found as follows:
For a second example:
m11
= 4 ¢ 1 ¡ 2 ¢ 16 = ¡36
(17)
m12
= ¡12 ¢ 1 ¡ 2 (¡1) = ¡10
(18)
m13
= ¡12 ¢ 16 ¡ 4 (¡1) = ¡188
(19)
m21
= 0 ¢ 1 ¡ (¡20) 16 = 320
(20)
m22
= ¡4 ¢ 1 ¡ (¡20) (¡1) = ¡24
(21)
m23
= ¡4 ¢ 16 ¡ 0 ¢ (¡1) = ¡64
(22)
m31
= 0 ¢ 2 ¡ (¡20) 4 = 80
(23)
m32
= ¡4 ¢ 2 ¡ (¡20) ¢ (¡12) = ¡248
(24)
m33
= ¡4 ¢ 4 ¡ 0 ¢ (¡12) = ¡16
(25)
0
-8.0000
B
A = @ 12.0000
-7.0000
hence:
0
-28.0000
B
M = @ 320.0000
80.0000
-10.0000
-24.0000
-248.0000
1
-188.0000
C
-64.0000 A
-16.0000
1
6.0000
C
5.0000 A
9.0000
-10.0000
-1.0000
2.0000
(27)
(28)
m11
= ¡1 ¢ 9 ¡ 5 ¢ 2 = ¡19
m12
= 12 ¢ 9 ¡ 5 (¡7) = 143
m13
= 12 ¢ 2 ¡ (¡1) (¡7) = 17
(30)
m21
= ¡10 ¢ 9 ¡ 6 ¢ 2 = ¡102
(31)
m22
= ¡8 ¢ 9 ¡ 6 ¢ (¡7) = ¡30
(32)
m23
= ¡8 ¢ 2 ¡ (¡10) ¢ (¡7) = ¡86
(33)
m31
= ¡10 ¢ 5 ¡ 6 ¢ (¡1) = ¡44
(34)
m32
= ¡8 ¢ 5 ¡ 6 ¢ 12 = ¡112
(35)
(29)
m33
= ¡8 ¢ (¡1) ¡ (¡10) 12 = 128
1
0
-19.0000
143.0000
17.0000
C
B
M = @ -102.0000 -30.0000
-86.0000 A
-44.0000 -112.0000 128.0000
(26)
2.2
(36)
(37)
Co-factors
Related to the minor is the concept of a co-factor. If the sum of the indices i
and j is an even number the value of the minor and the co-factor is the same. If
the sum of i and j is odd, the co-factor is the negative of the minor. Formally:
cij = (¡1)i+j mij
(38)
Thus for the matrix in 16 which has the matrix of minors in 26, the co-factor
matrix is given by:
0
-28.0000
B
C = @ -320.0000
80.0000
Richard Ti¢n
January 30, 2001
3
Richard Ti¢n
10.0000
-24.0000
248.0000
January 30, 2001
-188.0000
1
C
64.0000 A
-16.0000
(39)
4
and for the matrix in 27 which has the matrix of minors in 37 the co-factor
matrix is given by:
0
-19.0000
B
C = @ 102.0000
-44.0000
2.3
-143.0000
-30.0000
112.0000
1
17.0000
C
86.0000 A
128.0000
(40)
The Laplace expansion
The co-factors are useful in …nding the determinant of a higher order matrix.
The determinant of an nth order matrix may be evaluated as:
jAj =
n
X
aij cij
(41)
2.4
Evaluating a fourth order determinant
To evaluate a fourth order determinant is somewhat more involved since it
requires the application of the Laplace expansion at two levels. We will …nd the
determinant of:
1
0
3 6 2 1
C
B
B 0 4 0 3 C
C
(48)
A =B
B 2 8 9 7 C
A
@
3 1 2 8
To do this will take the Laplace expansion of the second row (for obvious reasons):
j=1
jAj = 0 ¢ c21 + 4 ¢ c22 + 0 ¢ c23 + 3 ¢ c24
which is termed the Laplace expansion of the ith row. Alternatively:
jAj =
n
X
aij cij
(42)
i=1
We therefore need to evaluate the co-factors c22 and c24 . We begin with c22 and
we start by deleting the second row and the …rst column so that the minor from
which c22 is obtained is found by evaluating the following determinant:
which is termed the Laplace expansion of the j th column, may be used.
For the matrix in 16 the Laplace expansion of the …rst row is given by:
jAj = ¡4 ¢ (¡28) + 0 ¢ 10 + (¡20) ¢ (¡188) = 3872
(49)
m22
(43)
¯
¯ 3 2 1
¯
¯
=¯ 2 9 7
¯
¯ 3 2 8
¯
¯
¯
¯
¯
¯
¯
(50)
which we will …nd by evaluating the Laplace expansion of the …rst row:
the expansion of the second row is given by:
jAj = (¡12) (¡320) + 4 ¢ (¡24) + 2 ¢ (64) = 3872
(51)
where the superscript is used to distinguish between these co-factors and those
in 49. The co-factors in 51 can be evaluated as follows, …rst obtaining the minors
and the Laplace expansion of the …rst column is given by:
jAj = (¡4) ¢ (¡28) + (¡12) ¢ (¡320) + (¡1) ¢ (80) = 3872
m22 = 3 ¢ c011 + 2 ¢ c012 + 1 ¢ c013
(44)
(45)
Since you can choose which row or column to expand, pick an easy one, in particular look for rows or columns with zeros in them. Notice that the expression
for the determinant in 8 is simply the Laplace expansion of the …rst row written
out in full since rewriting it slightly gives:
jAj = a11 (a22 a33 ¡ a23 a32 ) + a12 (a23 a31 ¡ a21 a33 )
+a13 (a21 a32 ¡ a22 a31 )
Richard Ti¢n
January 30, 2001
(46)
(47)
5
Richard Ti¢n
January 30, 2001
6
and then converting these to co-factors as appropriate:
¯
7 ¯¯
¯ = 9 ¢ 8 ¡ 7 ¢ 2 = 58
8 ¯
¯
¯ 9
¯
= ¯
¯ 2
m011
c011
= 58
¯
¯ 2
¯
= ¯
¯ 3
m012
c012
7
8
= 5
¯
¯ 2
¯
= ¯
¯ 3
m013
c13
¯
¯
¯
¯ = 2 ¢ 8 ¡ 7 ¢ 3 = ¡5
¯
¯
9 ¯¯
¯ = 2 ¢ 2 ¡ 9 ¢ 3 = ¡23
2 ¯
= ¡23
then converting these to co-factors:
(52)
m0031
(53)
c0011
(54)
m0032
(55)
c012
(56)
m0033
(57)
c13
using 51:
¯
¯ 6 2
¯
= ¯
¯ 8 9
¯
¯
¯
¯ = 6 ¢ 9 ¡ 2 ¢ 8 = 38
¯
= 38
¯
¯
¯ 3 2 ¯
¯
¯
= ¯
¯ = 3 ¢ 9 ¡ 2 ¢ 2 = 23
¯ 2 9 ¯
= ¡23
¯
¯
¯ 3 6 ¯
¯
¯
= ¯
¯ = 3 ¢ 8 ¡ 2 ¢ 6 = 12
¯ 2 8 ¯
= 12
(62)
(63)
(64)
(65)
(66)
(67)
using 61:
m22
c22
= 3 ¢ 58 + 2 ¢ 5 ¡ 1 ¢ 23 = 161
(58)
m22
= 161
(59)
c22
Now we turn to c24 . The minor from which c24 is obtained is found by
evaluating the following determinant:
m24
¯
¯ 3 6 2
¯
¯
=¯ 2 8 9
¯
¯ 3 1 2
¯
¯
¯
¯
¯
¯
¯
(68)
= 115
(69)
Using 59 and 69 in 49 gives:
jAj = 4 ¢ 161 + 3 ¢ 115 = 989
(60)
2.5
(70)
Properties of determinants
1. The determinant of A is equal to the determinant of A0 .
which we will …nd by evaluating the Laplace expansion of the …rst row:
m24 = 3 ¢ c0031 + 1 ¢ c0032 + 2 ¢ c0033
= 3 ¢ 38 + 1 ¢ (¡23) + 2 ¢ 12 = 115
(61)
The co-factors in 61 can be evaluated as above, …rst obtaining the minors and
In evaluating the determinant we can expand any row or column of the
matrix. Since A and A0 are identical except that the rows and columns
are transposed, the determinants of the two matrices will be identical.
2. The determinant of ¸A is equal to ¸n times the determinant of A where
A is an n-dimensional square matrix.
j¸Aj = ¸n jAj
(71)
This can be seen by examining equations 2 and 8. In each case, each
Richard Ti¢n
January 30, 2001
7
Richard Ti¢n
January 30, 2001
8
element of the matrix is multiplied by ¸:
The determinant of A2 can be calculated by expanding the second row:
(72)
j¸Aj = ¸a11 ¸a22 ¡ ¸a12 ¸a21
= ¸2 (a11 a22 ¡ a12 a21 ) = ¸2 jAj
jA2 j =
(73)
j¸Aj = ¸a11 ¸a22 ¸a33 ¡ ¸a11 ¸a23 ¸a32 + ¸a12 ¸a23 ¸a31 ¡ ¸a12 ¸a21 ¸a33
+¸a13 ¸a21 ¸a32 ¡ ¸a13 ¸a22 ¸a31
!
Ã
a11 a22 a33 ¡ a11 a23 a32 + a12 a23 a31 ¡ a12 a21 a33
= ¸3
+a13 (a21 a32 ¡ a22 a31 )
= ¸3 jAj
n
X
a1j c¤2j
(79)
j=1
(74)
the determinant in 79 is identical to that in 78 except that the co-factors
are di¤erent. However the minors used to …nd the co-factors are the same
in each case, the only di¤erence is that the row index for each will be 2
(75)
instead of 1. This means that the sum of the row and column indices will
increase by one in each case and the sign of the co-factor will change. In
e¤ect 79 is 78 multiplied through by minus one.
3. Multiplying a single row in a matrix by ¸ increases the value of the determinant by the same factor.
If we interchange the …rst and third rows:
0
a31
B
A3 = @ a21
a11
The reason is very similar to that in property 2 except that only one of
the elements in each term in 74 and 75 is multiplied by ¸.
4. Interchanging two rows in the matrix does not a¤ect the absolute value of
the determinant but its sign is reversed.
a32
a22
a12
1
a33
C
a23 A
a13
(80)
we can …nd the determinant by expanding the third row. The minors for
this row are given by:
Suppose we have the following matrix:
0
a11
B
A1 = @ a21
a31
a12
a22
a32
1
a13
C
a23 A
a33
m¤¤
11
(76)
a22
a12
a32
1
a23
C
a13 A
a33
(77)
We can compute the determinant of either matrix by expanding any row
(or column). Suppose we use the …rst row to expand A1 :
jA1 j =
Richard Ti¢n
n
X
a1j c1j
(78)
j=1
January 30, 2001
m¤¤
13
= a31 a22 ¡ a32 a21
(81)
= a31 a23 ¡ a33 a21
(82)
= a32 a23 ¡ a33 a22
(83)
noting that the minors used in evaluating 78 are:
and we swap the …rst and the second row:
0
a21
B
A2 = @ a11
a31
m¤¤
12
9
m11
= a21 a32 ¡ a22 a31
(84)
m12
= a21 a33 ¡ a23 a31
(85)
m13
= a22 a33 ¡ a23 a32
(86)
it can be seen that the expressions in 81 to 83 are the negative of those
in 84 to 86. The row indices of the elements of the matrix used in the
expansion now have two added to them and the pattern of sign changes in
…nding the co-factors is the same as with the row in its original position:
¤¤
¤¤
m¤¤
11 and m13 retain the same sign and m13 changes sign. Thus the sign of
each co-factor is again reversed but for slightly di¤erent reasons and this
again leads to a reversal in the sign of the determinant.
Richard Ti¢n
January 30, 2001
10
5. A matrix which has two identical rows or columns has a determinant of
zero.
Consider such a matrix and the e¤ect of interchanging two rows. As we
have just seen this must result in a change in the sign of the determinant.
However the new matrix is indistinguishable from the old one and it must
therefore have the same determinant. If both of these facts are to be true
the only value that the determinant can take is zero.
6. A matrix in which one row is a multiple of another row will have a determinant of zero.
except that the co-factors used in evaluating the determinant will be
the negative of those used in evaluating 87 because the row index of
the aij ’s in the expansion has increased by one. Property 5 shows
that this results in a value of zero, a result which would be una¤ected
by a change in the signs of the co-factors.
(b) Let us de…ne a matrix A¤ which is identical an original matrix A
except that the ith row of A¤ is obtained by taking the ith row of A
and adding k times the mth row of A. A typical element in the ith
row of A¤ is thus:
a¤ij = aij + kamj
Multiplying the row by the inverse of the relevant factor gives a matrix
with two identical rows which has a determinant of zero by property 5.
Therefore, by property 3 the determinant of the original matrix will be
zero.
The Laplace expansion of the ith row of A¤ is then:
jA¤ j =
7. The value of a determinant is not a¤ected by adding a multiple of one row
(column) to another row (column).
(a) Expansion of one row with the co-factors of another row1 results in
a value of zero:
n
X
(87)
amj cij = 0
In e¤ect this amounts to …nding the determinant of a matrix in which
row i is replaced with row j, giving a matrix in which two rows are
identical, except that the signs of the co-factors may be reversed.
For example if the original matrix is that in 76 evaluating L21 is
equivalent to evaluating the following determinant:
L¤21
1 This
a13
a13
a13
(90)
¯
¯
¯
¯
¯
¯
¯
jA¤ j =
=
n
X
(aij + kaim ) cij
(92)
j=1
n
X
aij cij + k
j=1
n
X
amj cij
(93)
j=1
The result in property 7a implies that the second term in 93 is zero.
Therefore:
jA¤ j =
(88)
(91)
and 90 can be written:
where Lmj indicates the expansion of row m with the co-factors of
row i.
a12
a12
a12
a¤ij c¤ij
j=1
c¤ij = cij
j=1
¯
¯ a11
¯
¯
= ¯ a11
¯
¯ a11
n
X
Since the elements of the ith row do not appear in the co-factors c¤ij
and all other elements in A¤ are identical to those in A:
To demonstrate this we proceed in two stages:
Lmi =
(89)
n
X
j=1
aij cij = jAj
(94)
This property can be useful in evaluating higher order determinants. For
is known as expansion by alien co-factors.
Richard Ti¢n
January 30, 2001
11
Richard Ti¢n
January 30, 2001
12
example2 if we are seeking the determinant of:
1
0
5
2 ¡2 1
C
B
B 12 6 ¡6 3 C
C
B
A =B
C
@ 7 ¡3 2 1 A
6
4
9 2
8. A matrix in which one row is a linear combination of other rows has a
determinant of zero.
(95)
aij = ¸akj + µamj
we can multiply the …rst row by ¡3 and subtract the result from row 2 to
produce:
0
5
2 ¡2 1
B
B
0
0
0
¤ B ¡3
A =B
@ 7 ¡3 2 ¡1
6
4
9
2
1
C
C
C
C
A
m21
(97)
the evaluation of which can be further simpli…ed by adding the …rst row
to the second row:
¯
¯
¯ ¡1 0 0 ¯
¯
¯
¯
¯
(98)
m21 = ¯ ¡3 2 ¡1 ¯
¯
¯
¯ 4 9 2 ¯
(103)
and let us derive a new matrix (in stages if you wish) in which the elements
of the ith row are given by:
a¤ij = aij ¡ ¸akj ¡ µamj
(96)
and the determinant of A¤ is the same as that of A. To evaluate jA¤ j we
only need to …nd the co-factor of a¤21 the minor for which is:
¯
¯
¯ 2 ¡2 1 ¯
¯
¯
¯
¯
= ¯ ¡3 2 ¡1 ¯
¯
¯
¯ 4
9
2 ¯
Suppose the elements of row i in a matrix are given by the following linear
combination of the elements in rows k and m of the matrix:
(104)
The new matrix has a row of zeros in the ith row and if we expand this row
it produces a determinant of zero. Since the new matrix is found from the
original matrix by subtracting a multiple of another row, the determinant
of the new matrix is identical to that of the original by property 7, and is
therefore zero.
References
Bailey, D. (1998). Mathematics in Economics, McGraw-Hill, Maidenhead, Berkshire.
expanding the …rst row:
m21
c21
= ¡1 (2 ¢ 2 + 1 ¢ 9)
(99)
= 13
(100)
= ¡13
(101)
therefore expanding row 2 of 96 gives:
jA¤ j = jAj = ¡3 ¢ ¡13 = 39
2 Taken
(102)
from (Bailey 1998, p.37).
Richard Ti¢n
January 30, 2001
13
Richard Ti¢n
January 30, 2001
14