APPENDIX
A
Matrix Algebra for
Structural Analysis
A.1 Basic Definitions and Types
of Matrices
With the accessibility of desk top computers, application of matrix algebra
for the analysis of structures has become widespread. Matrix algebra
provides an appropriate tool for this analysis, since it is relatively easy to
formulate the solution in a concise form and then perform the actual
matrix manipulations using a computer. For this reason it is important
that the structural engineer be familiar with the fundamental operations
of this type of mathematics.
Matrix. A matrix is a rectangular arrangement of numbers having m
rows and n columns.The numbers, which are called elements, are assembled
within brackets. For example, the A matrix is written as:
A = D
a11
a21
am1
a12
a22
Á
Á
a1n
a2n
am2
o
Á
amn
T
Such a matrix is said to have an order of m * n (m by n). Notice that the
first subscript for an element denotes its row position and the second
subscript denotes its column position. In general, then, aij is the element
located in the ith row and jth column.
Row Matrix. If the matrix consists only of elements in a single row,
it is called a row matrix. For example, a 1 * n row matrix is written as
A = [a1
a2
Á
an]
Here only a single subscript is used to denote an element, since the row
subscript is always understood to be equal to 1, that is, a1 = a11, a2 = a12,
and so on.
612
A.1
BASIC DEFINITIONS AND TYPES OF MATRICES
613
Column Matrix. A matrix with elements stacked in a single column
is called a column matrix. The m * 1 column matrix is
a1
a2
A = D T
o
am
Here the subscript notation symbolizes a1 = a11, a2 = a21, and so on.
Square Matrix. When the number of rows in a matrix equals the
number of columns, the matrix is referred to as a square matrix. An
n * n square matrix would be
A = D
a11
a21
an1
a12
a22
Á
Á
a1n
a2n
an2
o
Á
ann
T
Diagonal Matrix. When all the elements of a square matrix are
zero except along the main diagonal, running down from left to right, the
matrix is called a diagonal matrix. For example,
a11
A = C0
0
0
a22
0
0
0 S
a33
Unit or Identity Matrix. The unit or identity matrix is a diagonal
matrix with all the diagonal elements equal to unity. For example,
1
I = C0
0
0
1
0
0
0S
1
Symmetric Matrix. A square matrix is symmetric provided
aij = aji. For example,
3
A = C5
2
5
-1
4
2
4S
8
A
614
APPENDIX A
M AT R I X A L G E B R A
FOR
S T R U C T U R A L A N A LY S I S
A.2 Matrix Operations
Equality of Matrices. Matrices A and B are said to be equal if
they are of the same order and each of their corresponding elements are
equal, that is, aij = bij. For example, if
A = c
6
d
-3
2
4
B = c
6
d
-3
2
4
then A = B.
Addition and Subtraction of Matrices. Two matrices can be
added together or subtracted from one another if they are of the same
order. The result is obtained by adding or subtracting the corresponding
elements. For example, if
A = c
6
2
7
d
-1
1
3
15
d
3
B = c
-5
1
8
d
4
then
A + B = c
A
A - B = c
11
1
-1
d
-5
Multiplication by a Scalar. When a matrix is multiplied by a
scalar, each element of the matrix is multiplied by the scalar. For
example, if
A = c
4
6
1
d
-2
k = -6
then
kA = c
-24
-36
-6
d
12
Matrix Multiplication. Two matrices A and B can be multiplied
together only if they are conformable. This condition is satisfied if the
number of columns in A equals the number of rows in B. For example, if
A = c
a11
a21
a12
d
a22
B = c
b11
b21
b12
b22
b13
d
b23
(A–1)
then AB can be determined since A has two columns and B has two rows.
Notice, however, that BA is not possible. Why?
A.2
MATRIX OPERATIONS
615
If matrix A having an order of 1m * n2 is multiplied by matrix B
having an order of 1n * q2 it will yield a matrix C having an order of
1m * q2, that is,
A
=
B
1m * n21n * q2
C
1m * q2
The elements of matrix C are found using the elements aij of A and bij
of B as follows:
n
cij = a aikbkj
(A–2)
k=1
The methodology of this formula can be explained by a few simple
examples. Consider
2
A = c
-1
4
6
3
d
1
2
B = C6S
7
By inspection, the product C = AB is possible since the matrices are
conformable, that is, A has three columns and B has three rows. By
Eq. A–2, the multiplication will yield matrix C having two rows and one
column. The results are obtained as follows:
c11: Multiply the elements in the first row of A by corresponding elements
in the column of B and add the results; that is,
c11 = c1 = 2122 + 4162 + 3172 = 49
c21: Multiply the elements in the second row of A by corresponding
elements in the column of B and add the results; that is,
c21 = c2 = - 1122 + 6162 + 1172 = 41
Thus
C = c
49
d
41
A
616
APPENDIX A
M AT R I X A L G E B R A
FOR
S T R U C T U R A L A N A LY S I S
As a second example, consider
5
A = C 4
-2
3
1S
8
B = c
2
-3
7
d
4
Here again the product C = AB can be found since A has two columns
and B has two rows. The resulting matrix C will have three rows and two
columns. The elements are obtained as follows:
1first row of A times first column of B2
c11 = 5122 + 31-32 = 1
1first row of A times second column of B2
c12 = 5172 + 3142 = 47
1second row of A times first column of B2
c21 = 4122 + 11-32 = 5
1second row of A times second column of B2
c22 = 4172 + 1142 = 32
1third row of A times first column of B2
c31 = - 2122 + 81- 32 = - 28
1third row of A times second column of B2
c32 = - 2172 + 8142 = 18
The scheme for multiplication follows application of Eq. A–2. Thus,
1
C = C 5
-28
A
47
32 S
18
Notice also that BA does not exist, since written in this manner the
matrices are nonconformable.
The following rules apply to matrix multiplication.
1. In general the product of two matrices is not commutative:
AB Z BA
(A–3)
2. The distributive law is valid:
A1B + C2 = AB + AC
(A–4)
3. The associative law is valid:
A1BC2 = 1AB2C
(A–5)
Transposed Matrix. A matrix may be transposed by interchanging
its rows and columns. For example, if
a11
A = C a21
a31
a12
a22
a32
a13
a23 S
a33
B = [b1
b2
b3]
A.2
MATRIX OPERATIONS
617
Then
a11
A = C a12
a13
a21
a22
a23
T
a31
a32 S
a33
b1
B = C b2 S
b3
T
Notice that AB is nonconformable and so the product does not exist.
(A has three columns and B has one row.) Alternatively, multiplication
ABT is possible since here the matrices are conformable (A has three
columns and BT has three rows). The following properties for transposed
matrices hold:
1A + B2T = AT + BT
(A–6)
1kA2T = kAT
(A–7)
1AB2T = BTAT
(A–8)
This last identity will be illustrated by example. If
A = c
2
d
-3
6
1
B = c
4
2
3
d
5
A
Then, by Eq. A–8,
ac
6
1
3 T
4
db = c
5
3
2 4
dc
-3 2
ac
28
-2
c
2 6
dc
5 2
28
28 T
db = c
28
-12
-2
d
-12
-2
28
d = c
-12
28
-2
d
-12
28
28
1
d
-3
Matrix Partitioning. A matrix can be subdivided into submatrices
by partitioning. For example,
a11
A = C a21
a31
a12
a22
a32
a13
a23
a33
a14
A11
a24 S = c
A21
a34
A12
d
A22
Here the submatrices are
A11 = [a11]
A21 = c
a21
d
a31
A12 = [a12
a13
a14]
A22 = c
a23
a33
a24
d
a34
a22
a32
618
APPENDIX A
M AT R I X A L G E B R A
FOR
S T R U C T U R A L A N A LY S I S
The rules of matrix algebra apply to partitioned matrices provided the
partitioning is conformable. For example, corresponding submatrices of A
and B can be added or subtracted provided they have an equal number of
rows and columns. Likewise, matrix multiplication is possible provided
the respective number of columns and rows of both A and B and their
submatrices are equal. For instance, if
4
A = C -2
6
-1
-5 S
8
1
0
3
2
B = C0
7
-1
8S
4
then the product AB exists, since the number of columns of A equals the
number of rows of B (three). Likewise, the partitioned matrices are
conformable for multiplication since A is partitioned into two columns
and B is partitioned into two rows, that is,
AB = c
A12 B11
A B + A12B21
dc
d = c 11 11
d
A22 B21
A21B11 + A22B21
A11
A21
Multiplication of the submatrices yields
4 1 2 -1
8 4
dc
d = c
d
-2 0 0
8
-4 2
-1
-7
-4
= c
d[7
4] = c c
d
-5
-35 -20
2 -1
= [6
3] c
d = [12
18]
0
8
= [8][7
4] = [56
32]
A11B11 = c
A
A12B21
A21B11
A22B21
c
AB = D
8
-4
[12
4
-7
d + c
2
-35
18] + [56
-4
d
- 20
32]
1
T = C - 39
68
0
-18 S
50
A.3 Determinants
In the next section we will discuss how to invert a matrix. Since this
operation requires an evaluation of the determinant of the matrix, we will
now discuss some of the basic properties of determinants.
A determinant is a square array of numbers enclosed within vertical
bars. For example, an nth-order determinant, having n rows and n
columns, is
ƒAƒ = 4
a11
a21
an1
a12
a22
Á
Á
an2
o
Á
a1n
a2n 4
ann
(A–9)
A.3
DETERMINANTS
619
Evaluation of this determinant leads to a single numerical value which
can be determined using Laplace’s expansion. This method makes use of
the determinant’s minors and cofactors. Specifically, each element aij of a
determinant of nth order has a minor Mij which is a determinant of order
n - 1. This determinant (minor) remains when the ith row and jth column
in which the aij element is contained is canceled out. If the minor is
multiplied by 1-12i + j it is called the cofactor of aij and is denoted as
Cij = 1-12i + jMij
(A–10)
For example, consider the third-order determinant
a11
3 a21
a31
a12
a22
a32
a13
a23 3
a33
The cofactors for the elements in the first row are
C11 = 1-121 + 1 `
C12
C13
a22
a32
a
= 1-121 + 2 ` 21
a31
a
= 1-121 + 3 ` 21
a31
a23
a
` = ` 22
a33
a32
a23
a
` = - ` 21
a33
a31
a22
a
` = ` 21
a32
a31
a23
`
a33
a23
`
a33
a22
`
a32
A
Laplace’s expansion for a determinant of order n, Eq. A–9, states that
the numerical value represented by the determinant is equal to the sum
of the products of the elements of any row or column and their respective
cofactors, i.e.,
D = ai1Ci1 + ai2Ci2 + Á + ainCin
1i = 1, 2, Á , or n2
or
D = a1jC1j + a2jC2j + Á + anjCnj
1j = 1, 2, Á , or n2
(A–11)
For application, it is seen that due to the cofactors the number D is defined
in terms of n determinants (cofactors) of order n - 1 each. These
determinants can each be reevaluated using the same formula, whereby
one must then evaluate 1n - 12 determinants of order 1n - 22, and so
on. The process of evaluation continues until the remaining determinants
to be evaluated reduce to the second order, whereby the cofactors of the
elements are single elements of D. Consider, for example, the following
second-order determinant
D = `
3
-1
5
`
2
We can evaluate D along the top row of elements, which yields
D = 31 -121 + 1122 + 51-121 + 21- 12 = 11
Or, for example, using the second column of elements, we have
D = 51-121 + 21-12 + 21-122 + 2132 = 11
620
APPENDIX A
M AT R I X A L G E B R A
FOR
S T R U C T U R A L A N A LY S I S
Rather than using Eqs. A–11, it is perhaps easier to realize that the
evaluation of a second-order determinant can be performed by
multiplying the elements of the diagonal, from top left down to right, and
subtract from this the product of the elements from top right down to
left, i.e., follow the arrow,
3
-1
N
D = `
5
` = 3122 - 51-12 = 11
2
Consider next the third-order determinant
1
3
ƒDƒ =
4
-1
3
2
0
-1
63
2
Using Eq. A–11, we can evaluate ƒ D ƒ using the elements either along the
top row or the first column, that is
D = 1121- 121 + 1 `
2
0
6
4
` + 1321- 121 + 2 `
2
-1
6
4
` + 1- 121- 121 + 3 `
2
-1
2
`
0
= 114 - 02 - 318 + 62 - 110 + 22 = - 40
A
D = 11-121 + 1 `
2
0
6
3
` + 41- 122 + 1 `
2
0
-1
3
` + 1- 121- 123 + 1 `
2
2
-1
`
6
= 114 - 02 - 416 - 02 - 1118 + 22 = - 40
As an exercise try to evaluate ƒ D ƒ using the elements along the second row.
A.4 Inverse of a Matrix
Consider the following set of three linear equations:
a11 x1 + a12 x2 + a13 x3 = c1
a21 x1 + a22 x2 + a23 x3 = c2
a31 x1 + a32 x2 + a33 x3 = c3
which can be written in matrix form as
a11
C a21
a31
a12
a22
a32
a13
x1
c1
a23 S C x2 S = C c2 S
a33
x3
c3
Ax = C
(A–12)
(A–13)
A.4
INVERSE OF A MATRIX
621
One would think that a solution for x could be determined by dividing C
by A; however, division is not possible in matrix algebra. Instead, one
multiplies by the inverse of the matrix. The inverse of the matrix A is
another matrix of the same order and symbolically written as A-1. It has
the following property,
AA-1 = A-1A = I
where I is an identity matrix. Multiplying both sides of Eq. A–13 by A-1,
we obtain
A-1Ax = A-1C
Since A-1Ax = Ix = x, we have
x = A-1C
(A–14)
-1
Provided A can be obtained, a solution for x is possible.
For hand calculation the method used to formulate A-1 can be
developed using Cramer’s rule. The development will not be given here;
instead, only the results are given.* In this regard, the elements in the
matrices of Eq. A–14 can be written as
x = A-1C
x1
C11
1
C C12
C x2 S =
ƒAƒ
x3
C13
C21
C22
C23
C31 c1
C32 S C c2 S
C33 c3
(A–15)
Here ƒ A ƒ is an evaluation of the determinant of the coefficient matrix A,
which is determined using the Laplace expansion discussed in Sec. A.3.
The square matrix containing the cofactors Cij is called the adjoint matrix.
By comparison it can be seen that the inverse matrix A-1 is obtained from
A by first replacing each element aij by its cofactor Cij, then transposing
the resulting matrix, yielding the adjoint matrix, and finally multiplying the
adjoint matrix by 1> ƒ A ƒ .
To illustrate how to obtain A-1 numerically, we will consider the
solution of the following set of linear equations:
x1 - x2 + x3 = - 1
-x1 + x2 + x3 = - 1
x1 + 2x2 - 2x3 =
(A–16)
5
Here
1
A = C -1
1
-1
1
2
1
1S
-2
*See Kreyszig, E., Advanced Engineering Mathematics, John Wiley & Sons, Inc., New York.
A
622
APPENDIX A
M AT R I X A L G E B R A
FOR
S T R U C T U R A L A N A LY S I S
The cofactor matrix for A is
`
1
2
-1
C = F- `
2
-1
`
1
1
-1
` - `
-2
1
1
1
`
`
-2
1
1
1
` - `
1
-1
1
-1
`
`
-2
1
1
1
` - `
-2
1
1
1
`
`
1
-1
1
`
2
-1
`V
2
-1
`
1
Evaluating the determinants and taking the transpose, the adjoint
matrix is
-4
CT = C - 1
-3
0
-3
-3
-2
-2 S
0
Since
A
1
A = † -1
1
-1
1
2
1
1 † = -6
-2
The inverse of A is, therefore,
A-1 = -
-4
1
C -1
6
-3
0
-3
-3
-2
-2 S
0
Solution of Eqs. A–16 yields
x1
-4
1
C x2 S = - C - 1
6
x3
-3
0
-3
-3
-2
-1
- 2 S C -1 S
0
5
x1 = - 16[1-421 -12 + 01- 12 + 1-22152] = 1
x2 = - 16[1-121 -12 + 1- 321 -12 + 1- 22152] = 1
x3 = - 16[1-321 -12 + 1- 321 -12 + 102152] = - 1
Obviously, the numerical calculations are quite expanded for larger
sets of equations. For this reason, computers are used in structural
analysis to determine the inverse of matrices.
A.5
THE GAUSS METHOD FOR SOLVING SIMULTANEOUS EQUATIONS
623
A.5 The Gauss Method for Solving
Simultaneous Equations
When many simultaneous linear equations have to be solved, the Gauss
elimination method may be used because of its numerical efficiency.
Application of this method requires solving one of a set of n equations
for an unknown, say x1, in terms of all the other unknowns, x2, x3, Á , xn.
Substituting this so-called pivotal equation into the remaining equations
leaves a set of n - 1 equations with n - 1 unknowns. Repeating the
process by solving one of these equations for x2 in terms of the n - 2
remaining unknowns x3, x4, Á , xn forms the second pivotal equation.
This equation is then substituted into the other equations, leaving a set
of n - 3 equations with n - 3 unknowns. The process is repeated until
one is left with a pivotal equation having one unknown, which is then
solved. The other unknowns are then determined by successive back
substitution into the other pivotal equations. To improve the accuracy of
solution, when developing each pivotal equation one should always select
the equation of the set having the largest numerical coefficient for the
unknown one is trying to eliminate. The process will now be illustrated by
an example.
Solve the following set of equations using Gauss elimination:
-2x1 + 8x2 + 2x3 = 2
(A–17)
2x1 - x2 + x3 = 2
(A–18)
4x1 - 5x2 + 3x3 = 4
(A–19)
We will begin by eliminating x1. The largest coefficient of x1 is in
Eq. A–19; hence, we will take it to be the pivotal equation. Solving for x1,
we have
x1 = 1 + 1.25x2 - 0.75x3
(A–20)
Substituting into Eqs. A–17 and A–18 and simplifying yields
2.75x2 + 1.75x3 = 2
(A–21)
1.5x2 - 0.5x3 = 0
(A–22)
Next we eliminate x2. Choosing Eq. A–21 for the pivotal equation since
the coefficient of x2 is largest here, we have
x2 = 0.727 - 0.636x3
(A–23)
Substituting this equation into Eq. A–22 and simplifying yields the final
pivotal equation, which can be solved for x3. This yields x3 = 0.75.
Substituting this value into the pivotal Eq. A–23 gives x2 = 0.25. Finally,
from pivotal Eq. A–20 we get x1 = 0.75.
A