Notes On Elementary Operations on a Matrix - CBSE Class 12 Maths
Row and column operations on a matrix. These operations are particularly helpful when finding the inverse of a matrix, solving a system of linear equations and finding determinants. There are six operations available on a matrix. 1. Three on rows 2. Three on columns We have a matrix, A, with m rows and n columns. A = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Interchange of rows or columns = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$     $\underset{\to }{{\text{R}}_{\text{i}}\text{}\to \text{}{\text{R}}_{\text{j}}}$        $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$       Ex:         $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$        $\underset{\to }{{\text{R}}_{\text{2}}\text{}\to \text{}{\text{R}}_{\text{3}}}$              We have interchanged rows R2 and R3 Now, interchange the columns of a matrix. = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$       $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$                      Here, we have interchanged the elements of the columns Cp and Cq. Ex:   $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$                 We interchange the first and second columns. Multiplication of a row or column by a non-zero number. Ri → kRi, k ≠ 0 = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$     $\underset{\to }{{\text{R}}_{\text{i}}\text{}\to \text{}{\text{kR}}_{\text{i}}}$    $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{ka}}_{\text{i1}}& {\text{ka}}_{\text{i2}}& \text{⋯}& {\text{ka}}_{\text{ip}}& \text{⋯}& {\text{ka}}_{\text{iq}}& \text{⋯}& {\text{ka}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex:    $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$        $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{1}& \text{2/5}& \text{3/5}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$    Second row is multiplied with 1/5 Cp → lCp, l ≠ 0 = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$         $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{la}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{la}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{la}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{la}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{la}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex:   $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$                 Second column is multiplied by 4. Addition of elements of a row or column to the corresponding elements of any other row or column multiplied by a non-zero number Ri → Ri + kRj, k ≠ 0 A = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ First, we will multiply the jth row with the non-zero constant, k. Then we add these elements to the corresponding elements in the ith row A = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}\text{+ k}×{\text{a}}_{\text{j1}}& {{\text{a}}_{\text{i2}}\text{+k}×\text{a}}_{\text{j2}}& \text{⋯}& {{\text{a}}_{\text{ip}}\text{+k}×\text{a}}_{\text{jp}}& \text{⋯}& {{\text{a}}_{\text{iq}}\text{+k}×\text{a}}_{\text{jq}}& \text{⋯}& {{\text{a}}_{\text{in}}\text{+k}×\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex:  $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$   $\underset{\to }{{\text{R}}_{\text{2}}\text{}\to \text{}{\text{R}}_{\text{2}}\text{+ 3}{\text{R}}_{\text{3}}}$   $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5+3x(-2)}& \text{2+3(7)}& \text{3+3(1)}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$   =                          A similar type of operation over the columns of a matrix is Cp → Cp + lCq, l ≠ 0 A =$\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ A =  $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {{\text{a}}_{\text{1p}}\text{+l}×\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {{\text{a}}_{\text{2p}}\text{+l}×\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {{\text{a}}_{\text{ip}}\text{+l}×\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {{\text{a}}_{\text{jp}}\text{+l}×\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {{\text{a}}_{\text{mp}}\text{+l}×\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex: $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$         $\left[\begin{array}{ccc}\text{2+2x4}& \text{-1}& \text{4}\\ \text{5+2x3}& \text{2}& \text{3}\\ \text{-2+2x1}& \text{7}& \text{1}\end{array}\right]$   =

#### Summary

Row and column operations on a matrix. These operations are particularly helpful when finding the inverse of a matrix, solving a system of linear equations and finding determinants. There are six operations available on a matrix. 1. Three on rows 2. Three on columns We have a matrix, A, with m rows and n columns. A = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Interchange of rows or columns = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$     $\underset{\to }{{\text{R}}_{\text{i}}\text{}\to \text{}{\text{R}}_{\text{j}}}$        $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$       Ex:         $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$        $\underset{\to }{{\text{R}}_{\text{2}}\text{}\to \text{}{\text{R}}_{\text{3}}}$              We have interchanged rows R2 and R3 Now, interchange the columns of a matrix. = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$       $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$                      Here, we have interchanged the elements of the columns Cp and Cq. Ex:   $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$                 We interchange the first and second columns. Multiplication of a row or column by a non-zero number. Ri → kRi, k ≠ 0 = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$     $\underset{\to }{{\text{R}}_{\text{i}}\text{}\to \text{}{\text{kR}}_{\text{i}}}$    $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{ka}}_{\text{i1}}& {\text{ka}}_{\text{i2}}& \text{⋯}& {\text{ka}}_{\text{ip}}& \text{⋯}& {\text{ka}}_{\text{iq}}& \text{⋯}& {\text{ka}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex:    $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$        $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{1}& \text{2/5}& \text{3/5}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$    Second row is multiplied with 1/5 Cp → lCp, l ≠ 0 = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$         $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{la}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{la}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{la}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{la}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{la}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex:   $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$                 Second column is multiplied by 4. Addition of elements of a row or column to the corresponding elements of any other row or column multiplied by a non-zero number Ri → Ri + kRj, k ≠ 0 A = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ First, we will multiply the jth row with the non-zero constant, k. Then we add these elements to the corresponding elements in the ith row A = $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}\text{+ k}×{\text{a}}_{\text{j1}}& {{\text{a}}_{\text{i2}}\text{+k}×\text{a}}_{\text{j2}}& \text{⋯}& {{\text{a}}_{\text{ip}}\text{+k}×\text{a}}_{\text{jp}}& \text{⋯}& {{\text{a}}_{\text{iq}}\text{+k}×\text{a}}_{\text{jq}}& \text{⋯}& {{\text{a}}_{\text{in}}\text{+k}×\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex:  $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$   $\underset{\to }{{\text{R}}_{\text{2}}\text{}\to \text{}{\text{R}}_{\text{2}}\text{+ 3}{\text{R}}_{\text{3}}}$   $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5+3x(-2)}& \text{2+3(7)}& \text{3+3(1)}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$   =                          A similar type of operation over the columns of a matrix is Cp → Cp + lCq, l ≠ 0 A =$\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {\text{a}}_{\text{1p}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {\text{a}}_{\text{2p}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {\text{a}}_{\text{ip}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {\text{a}}_{\text{jp}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {\text{a}}_{\text{mp}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ A =  $\left[\begin{array}{cccccccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& \cdots & {{\text{a}}_{\text{1p}}\text{+l}×\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1q}}& \text{⋯}& {\text{a}}_{\text{1n}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& \text{⋯}& {{\text{a}}_{\text{2p}}\text{+l}×\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2q}}& \text{⋯}& {\text{a}}_{\text{2n}}\\ ⋮& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{i1}}& {\text{a}}_{\text{i2}}& \text{⋯}& {{\text{a}}_{\text{ip}}\text{+l}×\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{iq}}& \text{⋯}& {\text{a}}_{\text{in}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{j1}}& {\text{a}}_{\text{j2}}& \text{⋯}& {{\text{a}}_{\text{jp}}\text{+l}×\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jq}}& \text{⋯}& {\text{a}}_{\text{jn}}\\ \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}& \text{⋮}\\ {\text{a}}_{\text{m1}}& {\text{a}}_{\text{m2}}& \text{⋯}& {{\text{a}}_{\text{mp}}\text{+l}×\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mq}}& \text{⋯}& {\text{a}}_{\text{mn}}\end{array}\right]$ Ex: $\left[\begin{array}{ccc}\text{2}& \text{-1}& \text{4}\\ \text{5}& \text{2}& \text{3}\\ \text{-2}& \text{7}& \text{1}\end{array}\right]$         $\left[\begin{array}{ccc}\text{2+2x4}& \text{-1}& \text{4}\\ \text{5+2x3}& \text{2}& \text{3}\\ \text{-2+2x1}& \text{7}& \text{1}\end{array}\right]$   =

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