Find the rank of the matrix \[ A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} \]
Answer
This is the RREF of matrix A. The rank of a matrix is the maximum number of linearly independent rows, which in this case is 2.
Step 1 ：First, we perform Gaussian elimination on matrix A to put it in row-reduced echelon form (RREF).
Step 2 ：Subtract 4 times the first row from the second, and 7 times the first row from the third. This gives us \[ \begin{bmatrix} 1 & 2 & 3 \ 0 & -3 & -6 \ 0 & -6 & -12 \end{bmatrix} \].
Step 3 ：Next, divide the second row by -3, and add 2 times the second row to the third. This gives us \[ \begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 2 \ 0 & 0 & 0 \end{bmatrix} \].
Step 4 ：This is the RREF of matrix A. The rank of a matrix is the maximum number of linearly independent rows, which in this case is 2.
