Introduction to
Matrix Algebra
DEFINITION OF A MATRIX
A matrix is a rectangular array of
quantities arranged in rows and
columns. A matrix containing m
rows and n columns can be
expressed as
 A11 A12

A = [A] =  A21 A22

 
A
 m1 Am2
 A1n 
 A2n 

  
 Amn mxn
The quantities that form a matrix
are referred to as elements of the
matrix. Each element of the
matrix is identified with two
subscripts i and j to designate the
row and column locations,
respectively. Thus, the i,j element
(or coefficient) of [A] is expressed
as Aij with i = 1, 2, …, m and j = 1,
2, …, n for m rows and n
columns. This also defines the
size of [A], referred to as its order,
to be m x n.
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TYPES OF MATRICES
Row Matrix – If all the elements
of a matrix are arranged in a
single row (i.e., m = 1), then the
matrix is referred to as a row
matrix or row vector. A row
vector, denoted with angle
brackets, with n columns can be
expressed as
 r    r1 r2  rn 1xn
Notice that only one subscript is
used with the row vector to
identify the column location
since there is only one row.
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2
Column Matrix – If all the
elements of a matrix are
arranged in a single column (i.e.,
n = 1), then the matrix is referred
to as a column matrix or column
vector. A column vector, denoted
with braces, with m rows can be
expressed as
 c1 
c 
 
{c}   2 
  
c m 
Notice that only one
subscript is used
with the column
vector to identify
the row location
since there is only
one column.
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1
Square Matrix – Matrix with the
same number of rows and
columns, i.e., m = n. An nxn
square matrix is
 S11 S12  S1n 
S

21 S22  S2n 

[S] =
   
 
S

 n1 Sn2  Snn  nxn
Elements with the same subscript,
i.e., Sii for i = 1, 2, …, n; are
referred to as the diagonal
elements or coefficients. All other
coefficients Sij for i  j are termed
the off-diagonal elements or
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coefficients.
where all elements not shown are
zero.
Unit or Identity Matrix – diagonal
matrix with all diagonal elements
equal to 1 (i.e., Iii = 1 and Iij = 0 for i
 j) is called a unit or identity matrix
1

 1



[I] 
 


1 nxn

Null Matrix – When all the elements
of a matrix are zero [O] (i.e., Oij = 0
for i = 1, 2, …, m; j = 1, 2, …, n), the
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matrix is called a null matrix.
Symmetric Matrix – If the
elements of a square matrix are
symmetric (Sij = Sji) about the
main diagonal (Sii, i = 1, 2, …, n)
then the matrix is referred to as a
symmetric matrix.
Diagonal Matrix – If all the offdiagonal elements of a square
matrix are zero, i.e., Dij = 0 for i 
j; the matrix is referred to as a
diagonal matrix
 D11



D 22

[D]  




D nn  nxn

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MATRIX OPERATIONS
Equality – Two matrices [A] and [B]
are equal if they are of the same
order and if their corresponding
elements are identical, i.e., Aij = Bij
for i = 1, 2, …, m and j =1, 2, …, n.
Addition and Subtraction –
Addition or subtraction of two
matrices [A] and [B] is carried out
for two equal order matrices only by
adding or subtracting the
corresponding elements of the two
matrices, i.e.
Cij = Aij  Bij
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2
Multiplication by a Scalar – The
product of a scalar and a matrix is
obtained simply by multiplying
each matrix element by the scalar
c, i.e.
Cij = cAij
Multiplication of Matrices – The
multiplication of two matrices can
be carried out only if the number of
columns of the first (pre-multiplier)
matrix equals the number of rows
in the second (post-multiplier)
matrix. Such matrices are referred
to as conformable for
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multiplication.
Even when two matrices [A] and [B]
are of such orders that both matrix
products [A] [B] and [B] [A] can be
determined, the two products are
generally not equal:
[A] [B]  [B] [A]
i.e., matrix multiplication does not
satisfy the commutative law of
mathematics. Thus, it is necessary
to maintain proper sequential order
of matrices when computing matrix
products.
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For example,
[C]mxp = [A]mxn [B]nxp
or
Cij 
n
 Aik Bkj
k 1
 Ai1B1j  Ai2 B2 j   Ain Bnj
A common application of matrix
multiplication involves simultaneous equations
[A] {x} = {b}
which for a system or order three
is explicitly expressed as
 A11 A12 A13   x1   b1 
   
A
A 22 A 23   x 2   b 2 
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

   10
 A31 A32 A33   x 3   b3 
Matrix multiplication does satisfy
the associative and distributed
mathematical relationships
provided the matrices are
compatible, i.e.
[A] [B] [C] = ([A] [B]) [C]
= [A] ([B] [C])
[A] ([B] + [C]) = [A] [B] + [A] [C]
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3
Inverse of a Square Matrix – The
inverse of square matrix [S] is
represented as [S]-1 with the
elements of such magnitude that
the multiplication of the original [S]
by [S] -1 yields the identity matrix,
i.e.
[S] -1 [S] = [S] [S]-1 = [I]
Multiplication of [A] by a
compatible null matrix [O] results
in a null matrix and multiplication
of [A] by a compatible identity
matrix [I] results in matrix [A], i.e.
[A]mxn [O]nxp = [O]mxp
[O]pxm [A]mxn = [O]pxn
[A]mxn [I]nxn = [A]mxn
[I]mxm [A]mxn = [A]mxn
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Matrix inversion will be used in this
class for the solution of simultaneous equations
[S] {x} = {b}
{x} = [S]-1 {b}
An important property of matrix
inversion is that if [S] is symmetric,
then [S]-1 is also symmetric. The
simultaneous equations to be
solved in connection with the
displacement method of analysis
in this course will always be
symmetric.
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Your class notes include the
closed form inverse for a 2x2
matrix and a 3x3 matrix. Matrix
inversion is only defined for square
matrices and the order of the
inverse matrix is the same as the
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original matrix.
Transpose of a Matrix – The
transpose of a matrix is obtained by
interchanging its corresponding
rows and columns. Transpose of a
matrix is usually identified by the
superscript T on the matrix. For
example, consider [A]3x2, the
transpose of [A] is expressed as
[A]T and the elements of [A]T are
related to the elements of [A] as
 A11 A12 
[A]   A 21 A 22 


 A31 A32 
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4
A
[A]T   11
 A12
A 21
A 22
A31 
A32 
 A11 A12
A
A 22
[A]   21


A
 31 A32
For a symmetric matrix [S]:
[S] = [S]T ; sij = sji
The transpose of a matrix product
is defined as:
([A] [B]) T = [B] T [A] T
([A] [B] [C]) T = [C] T [B] T [A]T
Partitioning of Matrices –
Partitioning is a process by which
a matrix is subdivided into a
number of smaller matrices called
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submatrices. For example,
Matrix operations such as
addition, subtraction, and
multiplication can be performed
on partitioned matrices in the
same manner as described
previously by treating
submatrices as elements,
provided the matrices are
partitioned such that they are
conformable.
A13  A14 
A 23  A 24 

   
A33  A34 
[A]11  [A]12 
     


[A]21  [A]22 
A12
A
[A]11   11
 A 21 A 22
A 
[A]12   14 
 A 24 
[A]21   A31 A32
[A]22   A34 
A13 
A 23 
A33 
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GAUSS-JORDAN
ELIMINATION
Solution of Simultaneous
Equations – The Gauss-Jordan
elimination method is one of the
numerous techniques available to
solve simultaneous equations,
particularly for hand solution.
Consider the following three
symmetric equations:
4 x1 + 2 x2 + 0 x3 = 4
2 x1 + 8 x2 + 2 x3 = -4
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0 x1 + 2 x2 + 4 x3 = 0
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5
When applying the Gauss-Jordan
method, it is usually convenient to
write the coefficient matrix [A] and
the right hand side vector {b} as
submatrices of a partitioned
augmented matrix:
4 2 0 4 
 2 8 2 4 


 0 2 4 0 
Matrix Inversion – Gauss Jordan
elimination can also be used to
determine the inverse of a matrix.
You simply follow the GaussJordan process described above
with the augmented matrix equal to
the identity matrix.
For example, calculate the
inverse of the matrix used in the
Gauss-Jordan elimination
example above:
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The unknown vector {x} can be
calculated using [A]-1 as
Augmented Matrix =
4 2 0 1 0 0
2 8 2 0 1 0


 0 2 4 0 0 1 
 7 2 1 
1
[A]1   2 4 2 

24 
 1 2 7 
23
{x} = [A]-1 {b}
 x1 
 7 2 1   4 
  1 
 
2 4 2  4 
x 2  


 x  24  1 2 7   0 

 
 3
 36  3 
 24 2 


  24
  1
24
 12 1 
 24  2 
24
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