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 non-zero 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 the matrix is in row echelon form, the rank is the number of non-zero rows in the matrix.
For the matrix $M$, we can perform the following row operations:
- Add $\frac{1}{4}$ of row 1 to row 2:
$$\left[ {\begin{array}{*{20}{c}} { – 4}&1&{ – 1} \ 0&-\frac{3}{4}&-\frac{3}{4} \ 7&{ – 3}&1 \end{array}} \right]$$
- Add $\frac{7}{4}$ of row 1 to row 3:
$$\left[ {\begin{array}{*{20}{c}} { – 4}&1&{ – 1} \ 0&-\frac{3}{4}&-\frac{3}{4} \ 0&-\frac{11}{4}&-\frac{3}{4} \end{array}} \right]$$
- Swap row 2 with row 3:
$$\left[ {\begin{array}{*{20}{c}} { – 4}&1&{ – 1} \ 0&-\frac{11}{4}&-\frac{3}{4} \ 0&-\frac{3}{4}&-\frac{3}{4} \end{array}} \right]$$
*Subtract $\frac{3}{11}$ of row 2 from row 3:
$$\left[ {\begin{array}{*{20}{c}} { – 4}&1&{ – 1} \ 0&-\frac{11}{4}&-\frac{3}{4} \ 0&0&-\frac{3}{11} \end{array}} \right]$$
Therefore, the rank of the matrix $M$ is $\boxed{2}$.