JEE Main & Advanced Mathematics Determinants & Matrices Types of Matrices

Types of Matrices

Category : JEE Main & Advanced

(1) Row matrix : A matrix is said to be a row matrix or row vector if it has only one row and any number of columns.

Example :  [5  0  3] is a row matrix of order $1\times 3$ and [2] is a row matrix of order $1\times 1$.

(2) Column matrix : A matrix is said to be a column matrix or column vector if it has only one column and any number of rows.

Example : \left[ \begin{align} & \,\,\,2 \\ & \,\,\,3 \\ & -6 \\ \end{align} \right] is a column matrix of order $3\times 1$ and [2] is a column matrix of order $1\times 1$. Observe that [2] is both a row matrix as well as a column matrix.

(3) Singleton matrix : If in a matrix there is only one element then it is called singleton matrix.

Thus, $A={{[{{a}_{ij}}]}_{m\times n}}$is a singleton matrix, if $m=n=1$

Example : $[2],\text{ }[3],\text{ }[a],\text{ }[3]$ are singleton matrices.

(4) Null or zero matrix : If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by $O$. Thus $A={{[{{a}_{ij}}]}_{m\times n}}$is a zero matrix if ${{a}_{ij}}=0$for all $i$ and $j$.

Example : $[0],\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right],\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right],[0\,\,0]$ are all zero matrices, but of different orders.

(5) Square matrix : If number of rows and number of columns in a matrix are equal, then it is called a square matrix.

Thus $A={{[{{a}_{ij}}]}_{m\times n}}$is a square matrix if $m=n$.

Example : $\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right]$is a square matrix of order $3\times 3$.

(i) If $m\ne n$then matrix is called a rectangular matrix.

(ii) The elements of a square matrix A for which $i=j,i.e.\,\,{{a}_{11}},$ ${{a}_{22}},{{a}_{33}},....{{a}_{nn}}$are called diagonal elements and the line joining these elements is called the principal diagonal or leading diagonal of matrix A.

(6) Diagonal matrix : If all elements except the principal diagonal in a square matrix are zero, it is called a diagonal matrix. Thus a square matrix $A=[{{a}_{ij}}]$ is a diagonal matrix if $\Delta$when $\Delta =0$.

Example : $\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \\ \end{matrix} \right]$is a diagonal matrix of order $3\times 3$, which can be denoted by diag [2, 3, 4].

(7) Identity matrix : A square matrix in which elements in the main diagonal are all '1' and rest are all zero is called an identity matrix or unit matrix. Thus, the square matrix $A=[{{a}_{ij}}]$is an identity matrix, if {{a}_{ij}}=\left\{ \begin{align} & 1,\,\,\text{if}\,\,\,i=j \\ & 0,\,\,\text{if}\,\,i\ne j \\ \end{align} \right.

We denote the identity matrix of order $n$ by ${{I}_{n}}$.

Example : [1], $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\,,\left[ \,\,\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]$ are identity matrices of order 1, 2 and 3 respectively.

(8) Scalar matrix : A square matrix whose all non diagonal elements are zero and diagonal elements are equal is called a scalar matrix. Thus, if $A=[{{a}_{ij}}]$is a square matrix and {{a}_{ij}}=\left\{ \begin{align} & \alpha ,\,\text{if}\,\,i=j \\ & 0,\,\,\text{if}\,\,i\ne j \\ \end{align} \right., then A is a scalar matrix.

Unit matrix and null square matrices are also scalar matrices.

(9) Triangular matrix : A square matrix $[{{a}_{ij}}]$is said to be triangular matrix if each element above or below the principal diagonal is zero. It is of two types

(i) Upper triangular matrix : A square matrix $[{{a}_{ij}}]$is called the upper triangular matrix, if ${{a}_{ij}}=0$ when $i>j$.

Example : $\left[ \begin{matrix} 3 & 1 & 2 \\ 0 & 4 & 3 \\ 0 & 0 & 6 \\ \end{matrix} \right]$is an upper triangular matrix of order $3\times 3$.

(ii) Lower triangular matrix : A square matrix $[{{a}_{ij}}]$is called the lower triangular matrix, if ${{a}_{ij}}=0$ when$i<j$.

Example : $\left[ \begin{matrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 2 \\ \end{matrix} \right]$ is a lower triangular matrix of order $3\times 3$.

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