# Are three zeros always required in the third row for a matrix in echelon form?

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In summary, the conversation discusses the requirement for three zeros in the third row when finding the echelon form of a matrix. The expert explains that as long as the first non-zero term of each row is further to the right than the row above it and all identically zero rows are at the bottom, echelon form is achieved. They also mention the option of reduced echelon form, which would have additional zeros in certain positions. The expert concludes by giving an example of a matrix in echelon form but not reduced.
ver_mathstats
I need to find the echelon form of:

1 1 2 8
-1 -2 3 1
3 -7 4 10

and so far I have:

1 1 2 8
0 10 -50 -90
0 0 -52 -104

I was just wondering if I was required to put another zero in my third row. Am I always required to have three zeros in the third row? I'm assuming I do, but when I looked at the solution for this problem I found it to be:

1 1 2 8
0 1 -5 -9
0 0 1 2

However this solution does not have three zeros in the third row, when all of the other problems and solutions did have three zeros.

My apologies for the misaligned matrix, I am still getting used to Physics Forum.

Thank you.

no you don't need 3 zeroes in the 3rd row. as long as the first non zero term of every row is further to the right than the first non zero term in the row above it, and all identically zero rows are at the bottom, my impression is that echelon form is achieved. i sort of like reduced echelon form though, which would achieve also (in your case) zeroes in the (1,2) position, and the (1,3) and (2,3) positions, where the (n,m) entry is the one in the nth row and mth column.notice that any matrix consisting of only one row is already in echelon form, no matter how many or how few zeroes there are, if that helps.

note also the matrix whose rows are ( 1 1 1 1 1 1), (0 1 1 1 1 1) , (0 0 1 1 1 1) is in echelon form, but not reduced.

ver_mathstats
mathwonk said:
no you don't need 3 zeroes in the 3rd row. as long as the first non zero term of every row is further to the right than the first non zero term in the row above it, and all identically zero rows are at the bottom, my impression is that echelon form is achieved. i sort of like reduced echelon form though, which would achieve also (in your case) zeroes in the (1,2) position, and the (1,3) and (2,3) positions, where the (n,m) entry is the one in the nth row and mth column.notice that any matrix consisting of only one row is already in echelon form, no matter how many or how few zeroes there are, if that helps.

note also the matrix whose rows are ( 1 1 1 1 1 1), (0 1 1 1 1 1) , (0 0 1 1 1 1) is in echelon form, but not reduced.
Okay thank you very much, I understand it much better now and I can now see how I can reduce the solution I was given.

yes, all you can guarantee about a 3xn matrix with 3 rows in echelon form is that the 3rd row has at least 2 zeroes, since the entries below both previous pivots must be zero.

ver_mathstats

## What is the echelon form?

The echelon form is a matrix representation of a system of linear equations in which the leading coefficient of each row is to the right of the leading coefficient of the row above it, and any rows of all zeros are at the bottom. This form is useful in solving systems of equations and performing other matrix operations.

## Why is finding the echelon form important?

Finding the echelon form is important because it simplifies and organizes a system of linear equations, making it easier to solve and perform other operations. It also helps in determining the rank of a matrix, which is a key factor in determining whether a system of equations has a unique solution.

## What is the process for finding the echelon form?

The process for finding the echelon form involves using elementary row operations on a matrix to transform it into the desired form. These operations include swapping rows, multiplying a row by a constant, and adding a multiple of one row to another. By performing these operations, the matrix is gradually transformed into the echelon form.

## Can any matrix be transformed into echelon form?

Not all matrices can be transformed into echelon form. In order for a matrix to be transformed into echelon form, it must be a square matrix or have more rows than columns. Additionally, the matrix must have a nonzero pivot in each row, meaning that the leading coefficient of each row must be a nonzero number.

## What are the applications of the echelon form in science?

The echelon form has many applications in science, particularly in fields such as physics, engineering, and statistics. It is used in solving systems of equations, finding the inverse of a matrix, and performing other operations on matrices. It is also used in data analysis and modeling to determine relationships between variables and make predictions.

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