Different Types of Matrices – Definitions and Examples

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.

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