Rank of matrix of order $2 \times 2$ and $3 \times 3$

Question

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

Answer 1

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).

Answer 2

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 $(e)$?

Answer 3

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