Reduced Row Echelon Form and Hexadecimal Numbers

Algebra Level 4

\[ A = \left[ \begin{array}{c c c} 1 & 0 & 0 \\
0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array}\right] \quad, \quad B= \left[ \begin{array}{ c c c} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \\ \end{array}\right] \quad,\quad C= \left[ \begin{array}{c c c c} 1 & 0 & 2 & 5\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{array}\right], \\ D = \left[ \begin{array}{c c c} 1 & 0 & 0\\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array}\right] \quad,\quad E = \left[ \begin{array}{c c} 1 & 0\\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \\ \end{array}\right] \]

Which of the matrices above are in their reduced row echelon form?

Concatenate the letters of the matrices that are in their reduced row echelon form in alphabetical order and convert your answer from hexadecimal to decimal.

As an explicit example, if you think matrices \(A, B\) and \(C\) are in their reduced row echelon form, then your answer is \( ABC_{16} = 10 \times 16^{2} + 11 \times 16 + 12 = \boxed{2748} \).


You may find the rules for reduced row echelon form on this page to be helpful.
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