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Can we say that this matrix is in row reduced echelon form? \begin{bmatrix}1&0&0&3&1\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&0&0\end{bmatrix} I know that, it has leading numbers as 1 and other rows are zeros. Is there a rule to have columns which have all zeros need to occur the beginning? Your help is highly appreciated. Thank you.

Math Wonderer
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1 Answers1

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No, columns of 0s don't have to appear at the beginning. There is no such rule as that. You are more interested in the rows. The only two rules are

  1. the zero rows are at the bottom

  2. the leading coefficient of a nonzero row is to the right of the one above it, if they were placed in the same row

Your matrix satisfies these two and thus is in row echelon form. But to get it in reduced row echelon form, it must be that

  1. Any column that contains a pivot has 0 in all other entries.

  2. the pivots are all 1's

So, to put your matrix in rref you subtract row 3 from row 1

$$\begin{bmatrix}1&0&0&3&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&0&0\end{bmatrix}$$

Vons
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  • It is in row echelon form, but is not reduced. – Théophile Feb 19 '21 at 07:16
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    In row reduced Echelon form each column containing a leading 1 has zeros in all its other entries, which is not the case for the 5th column. For RREF the rules are:1. The first non-zero number in the first row (the leading entry) is the number 1. 2. The second row also starts with the number 1, which is further to the right than the leading entry in the first row. For every subsequent row, the number 1 must be further to the right. 3. The leading entry in each row must be the only non-zero number in its column. 5. Any non-zero rows are placed at the bottom of the matrix. – absolute0 Feb 19 '21 at 07:17
  • Okay good to know – Vons Feb 19 '21 at 07:21