Rank of matrix of order $2 \times 2$ and $3 \times 3$ How Can I calculate Rank of matrix Using echlon Method::
$(a)\;\; \begin{pmatrix}
1 & -1\\ 
2 & 3
\end{pmatrix}$
$(b)\;\; \begin{pmatrix}
2 & 1\\ 
7 & 4
\end{pmatrix}$
$(c)\;\; \begin{pmatrix}
2 & 1\\ 
4 & 2
\end{pmatrix}$
$(d)\;\; \begin{pmatrix}
2 & -3 & 3\\ 
2 & 2 & 3\\
3 & -2 & 2
\end{pmatrix}$
$(e)\;\; \begin{pmatrix}
1 & 2 & 3\\ 
3 & 6 & 9\\
1 & 2 & 3
\end{pmatrix}$
although I have a knowledge of Using Determinant Method to calculate rank of Given matrix.
But in exercise it is calculate using echlon form
plz explain me in detail
Thanks
 A: You need to reduce the matrices to row-echelon form to determine the rank of the matrices. This entails the use elementary row operations.
How many non-zero rows do you end with? The number of non-zero rows in a reduced row echelon square matrix is equal to the rank of the matrix.
Right away: can you see how the second row in $(c)$ is a multiple of the first row? Reducing to row echelon form  (subtract two times the entries of the first row to the second row) will give you a row of zeros $\implies$ rank is one.
Similarly, what do you notice about the second and third rows of matrix $(e)$?
A: Follow this link to find your answer
If you are left with any doubt after reading this this, feel free to discuss. 
A: Hint: Reduce the matrices to row echelon form. The number of non-zero rows will be the rank of the matrix (so long as you're working with square matrices). Alternately, the number of pivot columns (a pivot column of a row echelon form matrix is a column in which some row has its first non-zero entry) is the rank of the matrix (this works for non-square matrices, too).
