

Matrices
and Determinants 
Matrix definition 
Square matrices 
The identity matrix 
Vectors 
The transpose of a
matrix 





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 1is 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 the 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}.










Precalculus contents
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