Here you will what is matrix and definitions of different types of matrices with examples.
Let’s begin –
What is Matrix ?
A set of mn numbers (real or imaginary) arranged in the form of a rectangular array of m rows and n columns is called an m \(\times\) n matrix (to be read as m by n matrix).
A m by n matrix is usually written as
A = \(\begin{bmatrix}a_{11} & a_{12} & …… & a_{1n} \\ a_{21} & a_{22} & …… & a_{2n}\\ . & . & . \\ a_{m1} & a_{m2} & …… & a_{mn} \end{bmatrix}\)
Different Types of Matrices
All different types of matrices with examples are given below :
Row Matrix
A matrix having only one row is called a row matrix or a row-vector.
Example : A = [ 1 2 -1 2 ] is a row matrix of order \(1 \times 4\).
Column Matrix
A matrix having only one column is called a column matrix or a column vector.
Example : \(\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\) is a column matrix of order \(3 \times 1\).
Square Matrix
A matrix in which number of rows is equal to the number of columns, say n, is called a square matrix of order n.
Example : the matrix \(\begin{bmatrix} 2 & 1 & -1 \\ 3 & -2 & 5 \\ 1 & 5 & -3 \end{bmatrix}\) is a square matrix of order \(3 \times 3\) in which diagonal elements are 2, -2 and -3.
Diagonal Matrix
A square matrix A = \([a_{ij}]_{n\times n}\) is called a diagonal matrix if all the elements, except those in the leading diagonal are zero.
Example : the matrix \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}\) is a diagonal denoted by A = diag [1, 2, 3].
Scalar Matrix
A square matrix A = \([a_{ij}]_{n\times n}\) is called a scalar matrix if
(i) \(a_{ij}\) = 0 for all i \(\ne\) j and,
(ii) \(a_{ii}\) = c, for all i, where c \(\ne\) 0
In other words, a diagonal matrix in which all the diagonal elements are equal is called the scalar matrix.
Example : the matrix \(\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}\) is scalar martix of order 2.
Identity or Unit Matrix
A square matrix A = \([a_{ij}]_{n\times n}\) is called a identity or unit matrix if
(i) \(a_{ij}\) = 0 for all i \(\ne\) j and,
(ii) \(a_{ii}\) = 1, for all i
In other words, a diagonal matrix in which all the diagonal elements is unity is called the unit matrix.
The identity matrix of order n is denoted by \(I_n\).
Example : the matrix \(I_2\) = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) is identity matrix of order 2.
Null Matrix
A matrix in which al elements are zero is called a null or a zero matrix,
Example : the matrix \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\) is null matrix of order 2.
Upper Triangular Matrix
A square matrix A = \([a_{ij}]\) is called an upper triangular matrix if \(a_{ij}\) = 0 for all i > j.
Thus, in an upper triangular matrix, all elements below the main diagonal are zero.
Example : \(\begin{bmatrix} 1 & 3 & 4 \\ 0 & 4 & 5 \\ 0 & 0 & 7 \end{bmatrix}\) is a upper triangular matrix.
Lower Triangular Matrix
A square matrix A = \([a_{ij}]\) is called an lower triangular matrix if \(a_{ij}\) = 0 for all i < j.
Thus, in an lower triangular matrix, all elements above the main diagonal are zero.
Example : \(\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 1 & 6 & 5 \end{bmatrix}\) is a lower triangular matrix.
