A rectangular matrix is in echelon kind (or heat echelon form) if it has the complying with three properties: 1. All nonzero rows are above any rows of every zeros. 2. Every leading entry of a heat is in a obelisk to the ideal of the leading entry that the row above it. 3. Every entries in a column listed below a leading entry room zeros.

If a matrix in echelon type satisfies the following added conditions, then it is in diminished echelon form (or decreased row echelon form): 4. The leading entry in each nonzero heat is 1. 5. Each leading 1 is the only nonzero entry in its column.



Definition A pivot position in a procession A is a ar in A that coincides to a leading 1 in the decreased echelon form of A. A pivot pillar is a pillar of A that contains a pivot position.

Source: straight Algebra and Its Applications, David C. Lay

In the solution, $\beginbmatrix1 &4 &5 &-9 &7 \\0 &2 &4 &-6 &-6 \\0 &0 &0 &-5 &0 \\0 &0 &0 &0 &0 \\\endbmatrix \space \textare not in reduced echelon form\\$The author haven"t yet checked if the positions in 2 and also -5 accomplish the meaning of pivot position as soon as the echelon matrix is reduced to a decreased echelon form.I think if it i do not care the reduced echelon form, the columns above the pivot positions, the writer determined, deserve to be nonzeros and also the entries in pivot positions, author dermined, can be $0$ and or have nonzero entry to its left. For this reason author"s identify from the above matrix that leading 1, 2, -5 are pivot positions there is no reducing the echelon is wrong.

Is over there something I"m missing? Or doesn"t it have actually a chance of obtaining a various pivot position once the echelon type is adjusted into a reduced echelon form?

In a $m×n$ procession in echelon type of a direct system for some hopeful integers m, n, permit the leading entries $(■)$ have any nonzero value, and the starred entries $(☆)$ have any type of value consisting of zero.

Leading entries $■$s in $R_1$ and also $R_2$ in one echelon procession can come to be leading 1 in a decreased echelon matrix through dividing them by $■$, and the entrance ☆ in $R_1$ over $■$ in $R_2$ have the right to be $0$ by individually a many of $■$.

So $R_1$ and $R_2$ in a matrix in echelon type becomes as follows:$\beginarrayrcl R_1\space & <■ ☆\cdots ☆☆☆☆>\\R_2\space & <0 ■\cdots ☆☆☆☆>\endarray \qquad ~\beginarrayrcl R_1\space & <1 0\cdots ☆☆☆☆>\\R_2 &<0 1\cdots ☆☆☆☆> \endarray$

For all integers k with $2≤k$R_k\space$ $<0 \cdots 0 ■☆☆\cdots ☆>$$R_k+1$ $<0 \cdots 0 0 ■☆\cdots ☆>$.

Subtracting a lot of of top entry of $R_k+1$ native $R_k$ have the right to make the entry above leading $■$ in $R_k+1$ be zero, and the top $■$s in $R_k$, $R_k+1$ can be 1 through separating the rows by top entry $■$s.

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So the rows in echelon matrix come to be the complying with in lessened $m×n$ echelon matrix:$\beginarrayrcl R_k\space & <0 \cdots 0 ■☆☆\cdots ☆>\\ R_k+1 & <0 \cdots 0 0 ■☆\cdots ☆>\\ \endarray \qquad \beginarrayrcl R_k & <0 \cdots 0 1 0 ☆\cdots ☆>\\R_k+1 & <0 \cdots 0 0 1 ☆\cdots ☆>\\ \endarray$

Hence, it"s discovered that leading 1s in diminished echelon form of $m×n$ matrix of a straight system exchange mail to the locations of the leading non-zero worths in a $m×n$ procession in echelon form of the direct system.