The rank of a matrix is the number of linearly independent rows or columns in the matrix. To find the rank of a matrix, we can use Gaussian elimination.
In Gaussian elimination, we reduce the matrix to row echelon form. A row echelon form is a matrix in which all the rows below the main diagonal are zero, and the leading coefficient of each nonzero row is 1.
To reduce the matrix to row echelon form, we can use the following operations:
- Add or subtract a multiple of one row to another row.
- Multiply a row by a non-zero constant.
- Swap two rows.
Once we have reduced the matrix to row echelon form, the number of nonzero rows in the row echelon form is the rank of the original matrix.
In this case, the matrix can be reduced to the following row echelon form:
[\left( {\begin{array}{*{20}{c}} 1&0&-2&{ – 1} \ 0&1&2&1 \ 0&0&1&0 \end{array}} \right)]
Therefore, the rank of the matrix is $\boxed{3}$.
Option A is incorrect because the matrix has 3 linearly independent rows.
Option B is incorrect because the matrix has 4 rows, but the rank of the matrix is only 3.
Option C is incorrect because the matrix has 3 columns, but the rank of the matrix is only 3.
Option D is incorrect because the matrix has 4 columns, but the rank of the matrix is only 3.