452 Appendix
s(1, 2, 3) = 1, s(2, 1, 3) = −1, s(1, 1, 3) = 0.
(20)
Because s(j 1 , j 2 , j 3 ) suppresses terms with two equal indices, and
assigns others a sign depending on the order of the indices, it is
the same as the permutation symbol, ε j 1 j 2 j 3 (Eqn A.3.39).
The definition of the determinant gives the familiar result for
n = 2:
| A | = det
a
a
a
a
j
j
11
12
21
22
1
2
1
2
1
2
⎛
⎝
⎜
⎞
⎠
⎟ =
=
=
∑ ∑ s(j 1 , j 2 )a 1j 1 a 2j 2
= s(1, 1)a 11 a 21 + s(1, 2)a 11 a 22 + s(2, 1)a 12 a 21 + s(2, 2)a 12 a 22
= a 11 a 22 − a 12 a 21 ,
(21)
because s(1, 1) = s(2, 2) = 0, s(1, 2) = 1, and s(2, 1) = −1. For
a matrix with only one element, the determinant equals the
matrix element.
Among the properties of determinants that we will find
useful in solving systems of equations are:
• The determinant of a matrix equals that of its transpose,
| A | = | A T |.
• If two rows or columns of a matrix are interchanged, the
determinant has the same absolute value but changes
sign.
• If one row (or column) is multiplied by a constant, the
determinant is multiplied by that constant.
• If a multiple of one row (or column) is added to another
row (or column), the determinant is unchanged.
• If two rows or columns of a matrix are the same, the
determinant is zero.
Proving these properties is left for the problems.
A.4.3 Inverse
For an n × n square matrix A, the inverse matrix A −1 is defined
such that multiplication by the inverse gives the identity matrix
A −1 A = AA −1 = I.
(22)
A −1 can be written in terms of the cofactor matrix, C, whose
elements
c ij = (−1) i+j | A ij |
(23)
are formed from the determinants of A ij , an (n − 1) × (n − 1)
square matrix formed by deleting the i th row and j th column
from A. If | A | is not zero,
A −1 = C T / | A |.
(24)
For the familiar n = 2 case, see problem 7.
A matrix whose determinant is zero does not have an inverse,
and is called singular. Because the determinant of a matrix with
two equal rows or columns is zero, such a matrix is singular.
More generally, a matrix is singular if a row or column is a
linear combination of the others.
The inverse of the matrix product AB, if AB is nonsingular,
obeys
(AB) −1 = B −1 A −1 .
(25)
A matrix A whose transpose equals its inverse,
A −1 = A T ,
(26)
is called orthogonal. By extension, a matrix A with complex
elements is unitary if its adjoint and inverse are equal
A −1 = A + .
(27)
A.4.4 Systems of linear equations
A vector–matrix representation is often used for systems of
linear equations. In this formulation, a system of m equations
for n unknown variables x i ,
a 11 x 1 + a 12 x 2 . . . + a 1n x n = b 1
a 21 x 1 + a 22 x 2 . . . + a 2n x n = b 2
. . .
a m1 x 1 + a m 2 x 2 . . . + a mn x n = b m
(28)
is written in the form
j
n
=
∑
1
a ij x j = b i or Ax = b,
(29)
by defining the matrix of coefficients and column vectors for
the unknowns and right-hand side,
A
a
a
a
a
a
a
a
a
a
x
x
x
b
b
n
n
m
m
mn
n
. . .
. . .
.
.
. . .
.
.
.
. . .
.
.
.
. . .
.
. . .
.
.
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
=
11
12
1
21
22
2
1
2
1
2
1
x
b
2 2
.
.
.
.
b m
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟ ⎟
(30)
The coefficient matrix A is m × n, because there is one row for
each equation, and one column for each unknown.
The Ax = b form illustrates that whether a system of equations can be solved depends on the matrix A. A system of equations is called homogeneous in the special case that b = 0, and
inhomogeneous for all other cases in which b ≠ 0. We consider
here only systems where the number of unknowns and equations
are equal, so the coefficient matrix A is square. If A possesses an
inverse, both sides can be premultiplied by A −1 , and
A −1 Ax = A −1 b = Ix = x
(31)
s(1, 2, 3) = 1, s(2, 1, 3) = −1, s(1, 1, 3) = 0.
(20)
Because s(j 1 , j 2 , j 3 ) suppresses terms with two equal indices, and
assigns others a sign depending on the order of the indices, it is
the same as the permutation symbol, ε j 1 j 2 j 3 (Eqn A.3.39).
The definition of the determinant gives the familiar result for
n = 2:
| A | = det
a
a
a
a
j
j
11
12
21
22
1
2
1
2
1
2
⎛
⎝
⎜
⎞
⎠
⎟ =
=
=
∑ ∑ s(j 1 , j 2 )a 1j 1 a 2j 2
= s(1, 1)a 11 a 21 + s(1, 2)a 11 a 22 + s(2, 1)a 12 a 21 + s(2, 2)a 12 a 22
= a 11 a 22 − a 12 a 21 ,
(21)
because s(1, 1) = s(2, 2) = 0, s(1, 2) = 1, and s(2, 1) = −1. For
a matrix with only one element, the determinant equals the
matrix element.
Among the properties of determinants that we will find
useful in solving systems of equations are:
• The determinant of a matrix equals that of its transpose,
| A | = | A T |.
• If two rows or columns of a matrix are interchanged, the
determinant has the same absolute value but changes
sign.
• If one row (or column) is multiplied by a constant, the
determinant is multiplied by that constant.
• If a multiple of one row (or column) is added to another
row (or column), the determinant is unchanged.
• If two rows or columns of a matrix are the same, the
determinant is zero.
Proving these properties is left for the problems.
A.4.3 Inverse
For an n × n square matrix A, the inverse matrix A −1 is defined
such that multiplication by the inverse gives the identity matrix
A −1 A = AA −1 = I.
(22)
A −1 can be written in terms of the cofactor matrix, C, whose
elements
c ij = (−1) i+j | A ij |
(23)
are formed from the determinants of A ij , an (n − 1) × (n − 1)
square matrix formed by deleting the i th row and j th column
from A. If | A | is not zero,
A −1 = C T / | A |.
(24)
For the familiar n = 2 case, see problem 7.
A matrix whose determinant is zero does not have an inverse,
and is called singular. Because the determinant of a matrix with
two equal rows or columns is zero, such a matrix is singular.
More generally, a matrix is singular if a row or column is a
linear combination of the others.
The inverse of the matrix product AB, if AB is nonsingular,
obeys
(AB) −1 = B −1 A −1 .
(25)
A matrix A whose transpose equals its inverse,
A −1 = A T ,
(26)
is called orthogonal. By extension, a matrix A with complex
elements is unitary if its adjoint and inverse are equal
A −1 = A + .
(27)
A.4.4 Systems of linear equations
A vector–matrix representation is often used for systems of
linear equations. In this formulation, a system of m equations
for n unknown variables x i ,
a 11 x 1 + a 12 x 2 . . . + a 1n x n = b 1
a 21 x 1 + a 22 x 2 . . . + a 2n x n = b 2
. . .
a m1 x 1 + a m 2 x 2 . . . + a mn x n = b m
(28)
is written in the form
j
n
=
∑
1
a ij x j = b i or Ax = b,
(29)
by defining the matrix of coefficients and column vectors for
the unknowns and right-hand side,
A
a
a
a
a
a
a
a
a
a
x
x
x
b
b
n
n
m
m
mn
n
. . .
. . .
.
.
. . .
.
.
.
. . .
.
.
.
. . .
.
. . .
.
.
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
=
11
12
1
21
22
2
1
2
1
2
1
x
b
2 2
.
.
.
.
b m
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟ ⎟
(30)
The coefficient matrix A is m × n, because there is one row for
each equation, and one column for each unknown.
The Ax = b form illustrates that whether a system of equations can be solved depends on the matrix A. A system of equations is called homogeneous in the special case that b = 0, and
inhomogeneous for all other cases in which b ≠ 0. We consider
here only systems where the number of unknowns and equations
are equal, so the coefficient matrix A is square. If A possesses an
inverse, both sides can be premultiplied by A −1 , and
A −1 Ax = A −1 b = Ix = x
(31)