Question

Find the Rank of the matrix given by :-

$\sf A = \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right]$ ​

Answer 1

$\large\underline{\sf{Solution-}}$

Given matrix is

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right]$

Now, we have to find the rank of matrix A.

We know,

Rank of a matrix is defined as number of non - zero rows of a matrix when its reduced to Echleon form.

So, Let we reduce this matrix to Echleon form.

So, given matrix is

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_3 \: \to \: R_3 - R_2}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\4&5&6\\ 3&3&3\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_2 \: \to \: R_2 - R_1}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\3&3&3\\ 3&3&3\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_3 \: \to \: R_3 - R_2}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\3&3&3\\ 0&0&0\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_2 \: \to \: \frac{1}{3} R_2}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\1&1&1\\ 0&0&0\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_2 \: \to \: R_2 - R_1}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\0& - 1& - 2\\ 0&0&0\end{array}\right]$

So, number of non - zero rows = 2

So, it implies Rank of matrix A = 2

Answer 2

Answer:

$\large\underline{\sf{Solution-}}$

Given matrix is

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right]$

Now, we have to find the rank of matrix A.

We know,

Rank of a matrix is defined as number of non - zero rows of a matrix when its reduced to Echleon form.

So, Let we reduce this matrix to Echleon form.

So, given matrix is

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_3 \: \to \: R_3 - R_2}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\4&5&6\\ 3&3&3\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_2 \: \to \: R_2 - R_1}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\3&3&3\\ 3&3&3\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_3 \: \to \: R_3 - R_2}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\3&3&3\\ 0&0&0\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_2 \: \to \: \frac{1}{3} R_2}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\1&1&1\\ 0&0&0\end{array}\right]$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{ OP \: R_2 \: \to \: R_2 - R_1}}$

$\rm :\longmapsto\:A = \left[\begin{array}{ccc}1&2&3\\0& - 1& - 2\\ 0&0&0\end{array}\right]$

So, number of non - zero rows = 2

So, it implies Rank of matrix A = 2