Rank of Matrix

Rank of Matrix : Rank of Matrix is the number of linearly independent rows of the matrix. It is also the number of linearly independent columns of the matrix.

The rank of a matrix \(A\) equals the maximum number of linearly independent columns of \(A\). In terms of rows the rank of \(A\) equals the maximum number of linearly independent rows of \(A\).

Thus the rank of a \(A\) is a whole number \(\left( 0,1,2,3,….. \right)\) and it is denoted as \(rank(A)\) or \(\rho(A)\) . If \(A\) is a zero matrix then the \(rank(A) = 0\).

Echelon Form of a Matrix

Example (1)

Find Echelon form of the following matrix

\[\left( {\begin{array}{*{20}{c}}2&{ – 1}&{ – 1}\\{ – 1}&2&{ – 1}\\{ – 1}&{ – 1}&2\end{array}}\right)\]

Solution:

Let \(A =\left( {\begin{array}{*{20}{c}}2&{ – 1}&{ – 1}\\{ – 1}&2&{ – 1}\\{ – 1}&{ – 1}&2\end{array}}\right)\)

performing elementary row operations on \(A\)

Step 1: \(R_2\rightarrow R_2+(\frac{1}{2})R_1, R_3\rightarrow R_3+(\frac{1}{2})R_1\)

\[\Rightarrow A \sim \left( {\begin{array}{*{20}{c}}2&{ – 1}&{ – 1}\\0&{\frac{3}{2}}&{ – \frac{3}{2}}\\0&{ – \frac{3}{2}}&{\frac{3}{2}}\end{array}} \right)\]

Step 2: \( R_3\rightarrow R_3+R_2\)

\[\Rightarrow A \sim \left( {\begin{array}{*{20}{c}}2&{ – 1}&{ – 1}\\0&{\frac{3}{2}}&{ – \frac{3}{2}}\\0&0&0\end{array}} \right)\]

Step 3 : \( R_1\rightarrow (\frac{1}{2})R_1, R_2\rightarrow (\frac{2}{3})R_2\)

\[\Rightarrow A \sim \left( {\begin{array}{*{20}{c}}1&-{ \frac{1}{2}}&-{ \frac{1}{2}}\\0&1&-1\\0&0&0\end{array}} \right)\]

This is required Echelon form of \(A\).