Notes On Fundamentals of Matrices - CBSE Class 12 Maths
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. Matrices are denoted with capital letters. Ex:  A =  In this matrix, each number is an element. The horizontal lines of the elements are called rows. This matrix contains three rows The vertical lines of elements are called columns. This matrix contains three columns The order of a matrix corresponds to the number of rows and columns the matrix has. ${\left[\begin{array}{ccc}\text{2}& \text{6}& \text{9}\\ \text{-2}& \text{4}& \text{2}\\ \text{5}& \text{8}& \text{1}\end{array}\right]}_{\text{3x3}}$        ${\left[\begin{array}{ccc}\text{3}& \text{2}& \text{8}\\ \text{4}& \text{5}& \text{2}\end{array}\right]}_{\text{2x3}}$       ${\left[\begin{array}{cc}\text{2}& \text{4}\\ \text{6}& \text{7}\\ \text{1}& \text{6}\end{array}\right]}_{\text{3X2}}$                                             In the matrices shown, the first matrix is of order three by three which corresponds to three rows and three columns. The second matrix is of order two by three which corresponds to two rows and three columns. The third matrix is of order three by two. On generalising, any matrix can be considered as a rectangular array of elements with 'm' number of rows and 'n' number of columns. i.e. A = [aij]mxn  'i' represents the row and 'j' represents the column. 1 ≤ i ≤ m, 1 ≤ j ≤ n and i,j ∈ N Various types of matrices Coloumn matrix Column matrix: A matrix is said to be a column matrix if it has only one column. Ex: C =  $\left[\begin{array}{c}\text{5}\\ \sqrt{\text{2}}\\ \text{-1}\end{array}\right]$        The matrix of order three by one. The general representation of a column matrix having m rows is C = [aij]mx1 Row matrix Row matrix: A matrix is said to be a row matrix if it has only one row. Ex: R = [3  1  5] The order of the matrix is one by three. The general representation of a row matrix having n columns is C = [aij]1xn Square matrix Square matrix: A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Ex:  S =  ${\left[\begin{array}{ccc}\text{2}& \text{6}& \text{9}\\ \text{-2}& \text{4}& \text{2}\\ \text{5}& \text{8}& \text{1}\end{array}\right]}_{\text{3x3}}$   This matrix has three rows and three columns. General representation of a square matrix of order n by n is S = [aij]nxn Diagonal matrix Diagonal matrix: A square matrix is said to be a diagonal matrix if all its non diagonal elements are zero. Ex: D =  These elements are called the diagonal elements of the matrix. The general representation of a diagonal matrix is D = [dij],dij = 0 , i ≠ j Scalar matrix Scalar matrix: A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal. Ex: S =  A scalar matrix can be represented as S = [Sij], ${\text{S}}_{\text{ij}}\mathit{\text{}}=\left\{\begin{array}{ll}0& \hfill \phantom{\rule{10}{0ex}}i\ne j\\ k& \hfill \phantom{\rule{10}{0ex}}i=j\end{array}\right\$ Identity matrix Identity matrix is a special case of scalar matrix, with the diagonal elements equal to one. Identity matrix: A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. Ex: I = $\left[\begin{array}{ccc}\text{1}& \text{0}& \text{0}\\ \text{0}& \text{1}& \text{0}\\ \text{0}& \text{0}& \text{1}\end{array}\right]$ Identity matrices of order two by two and three by three are I = ${\left[\begin{array}{ccc}\text{1}& \text{0}& \text{0}\\ \text{0}& \text{1}& \text{0}\\ \text{0}& \text{0}& \text{1}\end{array}\right]}_{\text{3x3}}$  J = ${\left[\begin{array}{cc}\text{1}& \text{0}\\ \text{0}& \text{1}\end{array}\right]}_{\text{2x2}}$ General representation of identity matrix is I = [dij], dij = Zero matrix or Null matrix Zero matrix: A matrix is said to be zero matrix or null matrix if all its elements are zero. Ex: O = 1x1 O = ${\left[\begin{array}{cc}\text{0}& \text{0}\\ \text{0}& \text{0}\end{array}\right]}_{\text{2x2}}$    O =  ${\left[\begin{array}{ccc}\text{0}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{0}\end{array}\right]}_{\text{3x3}}$

#### Summary

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. Matrices are denoted with capital letters. Ex:  A =  In this matrix, each number is an element. The horizontal lines of the elements are called rows. This matrix contains three rows The vertical lines of elements are called columns. This matrix contains three columns The order of a matrix corresponds to the number of rows and columns the matrix has. ${\left[\begin{array}{ccc}\text{2}& \text{6}& \text{9}\\ \text{-2}& \text{4}& \text{2}\\ \text{5}& \text{8}& \text{1}\end{array}\right]}_{\text{3x3}}$        ${\left[\begin{array}{ccc}\text{3}& \text{2}& \text{8}\\ \text{4}& \text{5}& \text{2}\end{array}\right]}_{\text{2x3}}$       ${\left[\begin{array}{cc}\text{2}& \text{4}\\ \text{6}& \text{7}\\ \text{1}& \text{6}\end{array}\right]}_{\text{3X2}}$                                             In the matrices shown, the first matrix is of order three by three which corresponds to three rows and three columns. The second matrix is of order two by three which corresponds to two rows and three columns. The third matrix is of order three by two. On generalising, any matrix can be considered as a rectangular array of elements with 'm' number of rows and 'n' number of columns. i.e. A = [aij]mxn  'i' represents the row and 'j' represents the column. 1 ≤ i ≤ m, 1 ≤ j ≤ n and i,j ∈ N Various types of matrices Coloumn matrix Column matrix: A matrix is said to be a column matrix if it has only one column. Ex: C =  $\left[\begin{array}{c}\text{5}\\ \sqrt{\text{2}}\\ \text{-1}\end{array}\right]$        The matrix of order three by one. The general representation of a column matrix having m rows is C = [aij]mx1 Row matrix Row matrix: A matrix is said to be a row matrix if it has only one row. Ex: R = [3  1  5] The order of the matrix is one by three. The general representation of a row matrix having n columns is C = [aij]1xn Square matrix Square matrix: A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Ex:  S =  ${\left[\begin{array}{ccc}\text{2}& \text{6}& \text{9}\\ \text{-2}& \text{4}& \text{2}\\ \text{5}& \text{8}& \text{1}\end{array}\right]}_{\text{3x3}}$   This matrix has three rows and three columns. General representation of a square matrix of order n by n is S = [aij]nxn Diagonal matrix Diagonal matrix: A square matrix is said to be a diagonal matrix if all its non diagonal elements are zero. Ex: D =  These elements are called the diagonal elements of the matrix. The general representation of a diagonal matrix is D = [dij],dij = 0 , i ≠ j Scalar matrix Scalar matrix: A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal. Ex: S =  A scalar matrix can be represented as S = [Sij], ${\text{S}}_{\text{ij}}\mathit{\text{}}=\left\{\begin{array}{ll}0& \hfill \phantom{\rule{10}{0ex}}i\ne j\\ k& \hfill \phantom{\rule{10}{0ex}}i=j\end{array}\right\$ Identity matrix Identity matrix is a special case of scalar matrix, with the diagonal elements equal to one. Identity matrix: A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. Ex: I = $\left[\begin{array}{ccc}\text{1}& \text{0}& \text{0}\\ \text{0}& \text{1}& \text{0}\\ \text{0}& \text{0}& \text{1}\end{array}\right]$ Identity matrices of order two by two and three by three are I = ${\left[\begin{array}{ccc}\text{1}& \text{0}& \text{0}\\ \text{0}& \text{1}& \text{0}\\ \text{0}& \text{0}& \text{1}\end{array}\right]}_{\text{3x3}}$  J = ${\left[\begin{array}{cc}\text{1}& \text{0}\\ \text{0}& \text{1}\end{array}\right]}_{\text{2x2}}$ General representation of identity matrix is I = [dij], dij = Zero matrix or Null matrix Zero matrix: A matrix is said to be zero matrix or null matrix if all its elements are zero. Ex: O = 1x1 O = ${\left[\begin{array}{cc}\text{0}& \text{0}\\ \text{0}& \text{0}\end{array}\right]}_{\text{2x2}}$    O =  ${\left[\begin{array}{ccc}\text{0}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{0}\end{array}\right]}_{\text{3x3}}$

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