Find the rank of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 1 \\ 3 & 1 & 7 \end{bmatrix} \)

Step 1 ：Step 1: Perform the Gaussian elimination on the given matrix. Swap row 1 with row 2, and then subtract 2 times row 1 from row 2, and 3 times row 1 from row 3 to obtain \( R = \begin{bmatrix} 2 & 5 & 1 \\ 0 & -3 & 1 \\ 0 & -14 & 4 \end{bmatrix} \)

Step 2 ：Step 2: Continue with the Gaussian elimination. Multiply row 2 by -1/3, add 14/3 times row 2 to row 3 to obtain \( R = \begin{bmatrix} 2 & 5 & 1 \\ 0 & 1 & -1/3 \\ 0 & 0 & 2/3 \end{bmatrix} \)

Step 3 ：Step 3: Continue with the Gaussian elimination. Multiply row 3 by 3/2, subtract 5 times row 2 from row 1, and subtract 1/2 times row 3 from row 1 to obtain \( R = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & -1/3 \\ 0 & 0 & 1 \end{bmatrix} \)

Step 4 ：Step 4: The given matrix is now in row-echelon form, and we can see that it has 3 non-zero rows. So, the rank of the matrix A is 3.