Question

Find the rank of the matrix $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$.

Answer 1

The rank of the matrix is equal to the number of non-zero rows in the matrix after reducing it to the echelon form

Given matrix

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

$R_2\to R_2-R_1$

$A=\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 1& 1& 1\end{array}\right]$

$R_3\to R_3-R_1$

$A=\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Hence the non-zero row in the above matrix is 1.

Therefore , rank is 1.