< b="" />Rank of a matrix:
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.

< b="" />Solution:
To find the rank of the given matrix, we need to perform row operations to reduce the matrix to its row echelon form or reduced row echelon form.

< b="" />Step 1: Write the augmented matrix:
We write the given system of equations as an augmented matrix. The augmented matrix is obtained by writing the coefficients of the variables in the system of equations.

The augmented matrix for the given system of equations is:
[ 4 2 1 3 | 0 ]
[ 6 3 4 7 | 0 ]
[ 2 1 0 1 | 0 ]

< b="" />Step 2: Perform row operations:
We perform row operations to reduce the augmented matrix to its row echelon form or reduced row echelon form.

- Row 2 = Row 2 - 3/2 * Row 1
- Row 3 = Row 3 - 1/2 * Row 1

The updated augmented matrix is:
[ 4 2 1 3 | 0 ]
[ 0 0 7/2 7/2 | 0 ]
[ 0 0 -1/2 -1/2 | 0 ]

< b="" />Step 3: Continue row operations:
We continue performing row operations to further reduce the augmented matrix.

- Row 2 = Row 2 * 2/7
- Row 3 = Row 3 * (-2)

The updated augmented matrix is:
[ 4 2 1 3 | 0 ]
[ 0 0 1 1 | 0 ]
[ 0 0 1 1 | 0 ]

< b="" />Step 4: Final row operations:
We perform final row operations to obtain the row echelon form or reduced row echelon form.

- Row 1 = Row 1 - Row 3
- Row 2 = Row 2 - Row 3

The final augmented matrix is:
[ 4 2 0 2 | 0 ]
[ 0 0 0 0 | 0 ]
[ 0 0 1 1 | 0 ]

< b="" />Step 5: Count the number of non-zero rows:
We count the number of non-zero rows in the row echelon form or reduced row echelon form of the augmented matrix.

In this case, there are 2 non-zero rows.

< b="" />Step 6: Conclusion:
The rank of the matrix is the number of non-zero rows in the row echelon form or reduced row echelon form of the augmented matrix.

Therefore, the rank of the given matrix is 2.