50. Which of the matrices [} 1&0&6&3 0&1&0&-2 0&0&1&4 ] given below is the reduced row-echclon form of the following matrix? A. [} 0&0&1&(1)/(2) 0&1&0&(3)/(4) 0&0&0&1 ] C. [} 1&0&0&(1)/(2) 0&1&0&(3)/(4) 0&0&1&0 ] B. [} 1&0&0&(3)/(4) 0&1&0&(1)/(2) 0&0&1&0 ] D. [} 1&0&0&0 0&1&0&(3)/(4) 0&0&1&(1)/(2) ]

To determine which matrix is the reduced row-echelon form (RREF) of the given matrix, we need to perform row reduction on the given matrix and compare it with the options provided.<br /><br />The given matrix is:<br />$[\begin{matrix} 1&0&6&3\\ 0&1&0&-2\\ 0&0&1&4\end{matrix} ]$<br /><br />Let's perform row reduction on this matrix:<br /><br />1. The first row already has a leading 1 in the first column, so we don't need to do anything to it.<br /><br />2. The second row already has a leading 1 in the second column, so we don't need to do anything to it.<br /><br />3. The third row already has a leading 1 in the third column, so we don't need to do anything to it.<br /><br />Now, let's compare the given matrix with the options provided:<br /><br />A. $[\begin{matrix} 0&0&1&\frac {1}{2}\\ 0&1&0&\frac {3}{4}\\ 0&0&0&1\end{matrix} ]$<br />B. $[\begin{matrix} 1&0&0&\frac {3}{4}\\ 0&1&0&\frac {1}{2}\\ 0&0&1&0\end{matrix} ]$<br />C. $[\begin{matrix} 1&0&0&\frac {1}{2}\\ 0&1&0&\frac {3}{4}\\ 0&0&1&0\end{matrix} ]$<br />D. $[\begin{matrix} 1&0&0&0\\ 0&1&0&\frac {3}{4}\\ 0&0&1&\frac {1}{2}\end{matrix} ]$<br /><br />Comparing the given matrix with the options, we can see that option C matches the given matrix after performing row reduction. Therefore, the correct answer is option C.