

Matrices and Determinants 
Matrix definition 
Square matrices 
The identity matrix 
Vectors 
The transpose of a
matrix 
Basic
matrix operations 
Matrix addition and
subtraction 
Scalar
multiplication of a matrix 
Matrix multiplication 
The
matrixvector product






Matrices and
Determinants 
Matrix definition 
A
matrix is an ordered rectangular array of elements or entries (components)
arranged in rows and columns
denoted by
bold
uppercase letters and indicated by brackets. 
A
matrix with m
rows and n
columns is called an mbyn
matrix that has an order (or dimension) of m
´
n 

The
element denoted a_{ij}
lies at the intersection of the i^{th}
row and the j^{th}
column, as is shown above. 

Square
matrices 
A
matrix with same number of rows and columns (m
= n)
is a square
matrix of order n. 
The
main diagonal of a square matrix is
the diagonal that runs from the top left corner to the bottom
right corner. 
A
square matrix with all components zero except those on the main
diagonal is called the diagonal matrix. 
The
identity matrix 
An
n
´
n
diagonal matrix with all nonzero elements equal to 1 is the identity
matrix of order n. 
An
upper triangular matrix is a square
matrix in which all elements underneath the main diagonal are
zeros. 
A
lower triangular matrix is a square
matrix in which all elements above the main diagonal are zeros. 
Square
matrices for which a_{ij}
= a_{ji}
are called symmetric about the main
diagonal. 
Examples
of square matrices 

The
diagonal matrix of order 4, 

The
identity matrix of order 3, 








The
upper triangular matrix of order 4, 

The
lower triangular matrix of order 4, 







The
symmetric matrix of order 5, 


The
elements of the main diagonal are in bold. 

A
matrix whose all elements are zero is called the null
matrix, written 0. 

Vectors 
A
matrix with only one row or one column is called a vector. 
A
matrix with one row and n
columns is called a row vector. 
A
matrix with one column and m
rows is called a column vector. 

Transposition 
The
transpose is the matrix derived from a given matrix by
interchanging the rows and columns. Thus, the
transpose of a matrix A
of order
m
´
n
is another matrix
denoted A^{T}
of order n
´
m. 
Therefore,
to transpose means to interchange the rows and columns of a
matrix, that
is, 
A^{T}
= [a_{ji}],
where j
is the column and i
is the row of matrix A
= [a_{ij}]. 
Example:
Given a matrix A
of order 3 ´
4, find the transpose A^{T}.



Basic
matrix
operations 
Matrix addition and
subtraction 
Matrices
can be added or subtracted only if they have the same
dimensions. 
Addition 
The
elements of the sum of the two matrices with the same
dimensions, A
and B,
are equal to the sums of the
corresponding elements a_{ij}
+ b_{ij}
that is, 
A_{}
+ B
= [a_{ij}
+ b_{ij}]. 
Subtraction 
The
elements of the difference of the two matrices with the same
dimensions, A
and B,
are equal to the differences
of the corresponding elements a_{ij}

b_{ij}
that is, 
A

B
= [a_{ij}

b_{ij}]. 
Example:
Given are matrices, A
and B, find the
sum A
+ B
and the difference A

B.



and 




then, 




and 




Scalar
multiplication of a matrix 
Multiplication
of a matrix A
by a scalar c
is defined as 
c
· A = [c · a_{ij}], 
that
is, each element of the matrix is multiplied by c.
Therefore, if c
= 0, the result is the null
matrix. 
Example:
Given the matrix 

find the
product 
2 · A. 




Matrix
multiplication 
Multiplication
of two matrices is defined only if the number of columns of the
first matrix equals the number of rows
of the second. Matrices that satisfy this condition are said to
be conformable. 
If
A
= [a_{ij}]
is a matrix of order m
´
n
and B
= [b_{jk}]
is a matrix of order n
´
p,
then the product 
C
= AB 
where
C
= [c_{ik}]
is the m
´
p
matrix of which the ik^{th}
entries are defined by the formula 

Therefore, the
ik^{th}
entry of the product equals the inner product of the components
of the i^{th}
row of the first matrix
with the components of the k^{th
}column of the
second matrix. 
Example:
Given are conformable matrices, A
and B, find the
product AB.



and 



then, 



or 




The
matrixvector product 
If
A = [a_{ij}]
is an m
´
n
matrix, x =
{x_{j}}
a column vector of order n,
and y = {y_{i}}
a column vector of order m 
then,
y = Ax
is the matrixvector product, 
where 



The
column dimension of the matrix A
must equal the row dimension of the vector x
to obey the product definition. 
Example:
Given is the matrix A
of order 2
´
3
and the column vector x
of order 3, find the
product Ax.



and 



then, 




If A is
a square matrix of order n
then,

AI =
IA = A 
where
I is the
identity
matrix of order n. 








Matrices, determinants and systems of linear equations
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