JEE Main & Advanced Mathematics Determinants & Matrices Types of Matrices

(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 \[3\times 3\].