2. Reduce each of the following matrices into Reduced Row Echelon Form. a. (} 1&2&1&2 2&1&2&1 ) b. (} 1&2&1 2&1&2 1&3&1 ) c. (} 1&2&3&1 4&5&6&2 7&8&9&3 )

a. To reduce the matrix $(\begin{matrix} 1&2&1&2\\ 2&1&2&1\end{matrix} )$ into Reduced Row Echelon Form (RREF), we can perform the following row operations:<br /><br />1. Subtract 2 times the first row from the second row to eliminate the 2 in the first column of the second row:<br />$(\begin{matrix} 1&2&1&2\\ 0&-3&0&-3\end{matrix} )$<br /><br />2. Divide the second row by -3 to make the leading coefficient 1:<br />$(\begin{matrix} 1&2&1&2\\ 0&1&0&1\end{matrix} )$<br /><br />3. Subtract 2 times the second row from the first row to eliminate the 2 in the second column of the first row:<br />$(\begin{matrix} 1&0&1&0\\ 0&1&0&1\end{matrix} )$<br /><br />The matrix is now in RREF.<br /><br />b. To reduce the matrix $(\begin{matrix} 1&2&1\\ 2&1&2\\ 1&3&1\end{matrix} )$ into RREF, we can perform the following row operations:<br /><br />1. Subtract 2 times the first row from the second row to eliminate the 2 in the first column of the second row:<br />$(\begin{matrix} 1&2&1\\ 0&-3&0\\ 1&3&1\end{matrix} )$<br /><br />2. Subtract the first row from the third row to eliminate the 1 in the first column of the third row:<br />$(\begin{matrix} 1&2&1\\ 0&-3&0\\ 0&1&0\end{matrix} )$<br /><br />3. Divide the second row by -3 to make the leading coefficient 1:<br />$(\begin{matrix} 1&2&1\\ 0&1&0\\ 0&1&0\end{matrix} )$<br /><br />4. Subtract the second row from the third row to eliminate the 1 in the second column of the third row:<br />$(\begin{matrix} 1&2&1\\ 0&1&0\\ 0&0&0\end{matrix} )$<br /><br />The matrix is now in RREF.<br /><br />c. To reduce the matrix $(\begin{matrix} 1&2&3&1\\ 4&5&6&2\\ 7&8&9&3\end{matrix} )$ into RREF, we can perform the following row operations:<br /><br />1. Subtract 4 times the first row from the second row to eliminate the 4 in the first column of the second row:<br />$(\begin{matrix} 1&2&3&1\\ 0&-3&-6&-2\\ 7&8&9&3\end{matrix} )$<br /><br />2. Subtract 7 times the first row from the third row to eliminate the 7 in the first column of the third row:<br />$(\begin{matrix} 1&2&3&1\\ 0&-3&-6&-2\\ 0&-6&-12&-4\end{matrix} )$<br /><br />3. Divide the second row by -3 to make the leading coefficient 1:<br />$(\begin{matrix} 1&2&3&1\\ 0&1&2&\frac{2}{3}\\ 0&-6&-12&-4\end{matrix} )$<br /><br />4. Add 6 times the second row to the third row to eliminate the -6 in the second column of the third row:<br />$(\begin{matrix} 1&2&3&1\\ 0&1&2&\frac{2}{3}\\ 0&0&0&0\end{matrix} )$<br /><br />The matrix is now in RREF.