Problem
\[ \left[\begin{array}{cc:c} 2 & 16 & -4 \\ 6 & 52 & -12 \end{array}\right] \stackrel{2 \cdot R_{1} \rightarrow R_{1}}{\longrightarrow}\left[\begin{array}{cc:c} 1 & 8 & -2 \\ 6 & 52 & -12 \end{array}\right] \] Step 2: \[ \left[\begin{array}{cc:c} 1 & 8 & -2 \\ 6 & 52 & -12 \end{array}\right] \stackrel{\text { [] } R_{1}+R_{2} \rightarrow R_{2}}{\longrightarrow}\left[\begin{array}{cc:c} 1 & 8 & -2 \\ 0 & 4 & 0 \end{array}\right] \] Step 3: \[ \left[\begin{array}{cc:c} 1 & 8 & -2 \\ 0 & 4 & 0 \end{array}\right] \stackrel{\left[\cdot R_{2} \rightarrow R_{2}\right.}{\longrightarrow}\left[\begin{array}{cc:c} 1 & 8 & -2 \\ 0 & 1 & 0 \end{array}\right] \] Step 4: Enter the coefficient for the row operations and the missing entries in the resulting matrix. \[ \left[\begin{array}{cc:c} 1 & 8 & -2 \\ 0 & 1 & 0 \end{array}\right] \stackrel{\left[\cdot R_{2}+R_{1} \rightarrow R_{1}\right.}{\longrightarrow}\left[\begin{array}{ll:l} 1 & 0 & \square \\ 0 & 1 & \square \end{array}\right] \]
Solution
Step 1 ：Given the matrix \[\left[\begin{array}{cc:c} 2 & 16 & -4 \\ 6 & 52 & -12 \end{array}\right]\], we first multiply the first row by 2 to get \[\left[\begin{array}{cc:c} 1 & 8 & -2 \\ 6 & 52 & -12 \end{array}\right]\].
Step 2 ：Next, we add the first row to the second row to get \[\left[\begin{array}{cc:c} 1 & 8 & -2 \\ 0 & 4 & 0 \end{array}\right]\].
Step 3 ：We then divide the second row by 4 to get \[\left[\begin{array}{cc:c} 1 & 8 & -2 \\ 0 & 1 & 0 \end{array}\right]\].
Step 4 ：Finally, we subtract 8 times the second row from the first row to get \[\left[\begin{array}{ll:l} 1 & 0 & \square \\ 0 & 1 & \square \end{array}\right]\].
Step 5 ：The missing entries in the resulting matrix are \(\boxed{-2}\) and \(\boxed{0}\).
