Use row operations to change the matrix to reduced form. \[ \left[\begin{array}{lll|l} 1 & 1 & 1 & 17 \\ 5 & 6 & 9 & 30 \end{array}\right] \] \[ \left[\begin{array}{lll|l} 1 & 1 & 1 & 17 \\ 5 & 6 & 9 & 30 \end{array}\right] \sim\left[\begin{array}{l} \square \square \square \\ \square \square \square \\ \square \square \end{array}\right] \]
Solution
Step 1 ：Subtract 5 times the first row from the second row to create a zero in the second row, first column. This gives us: \[\left[\begin{array}{lll|l} 1 & 1 & 1 & 17 \\ 0 & 1 & 4 & -55 \end{array}\right]\]
Step 2 ：Subtract the second row from the first row to create a zero in the first row, second column. This gives us: \[\left[\begin{array}{lll|l} 1 & 0 & -3 & 72 \\ 0 & 1 & 4 & -55 \end{array}\right]\]
Step 3 ：Add 3 times the second row to the first row to create a zero in the first row, third column. This gives us the reduced form of the matrix: \[\left[\begin{array}{lll|l} 1 & 0 & 0 & -93 \\ 0 & 1 & 4 & -55 \end{array}\right]\]
Step 4 ：So, the reduced form of the given matrix is: \[\boxed{\left[\begin{array}{lll|l} 1 & 0 & 0 & -93 \\ 0 & 1 & 4 & -55 \end{array}\right]}\]
