Quick easy question about Row echelon form

In summary, the rank of a matrix is the number of dimensions less than the number of rows in the matrix. If you have a matrix with n rows and m dimensions, the rank is n-1. A system of equations having that matrix as coefficient matrix will not have a unique solution- either no solution or an infinite number of solutions.
  • #1
dalarev
99
0
So google has yielded no good results. When I "transform" a matrix to row echelon form, not reduced row echelon form (leading entries are not necessarily 1), what does it mean if my last row is all 0's? Another thing, when setting up the equations as a matrix, do I include the solutions of the equations in there?
I did them with Cramer's rule and did not need the solution in the matrix. Thanks for the help.
 
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  • #2
It means that the rank of your matrix is one less than the dimension of the matrix. It also means that a system of equations having that matrix as coefficient matrix will not have a unique solution- either no solution or an infinite number of solutions.

Whether you include the "solutions of the equations" (by which I think you mean the right-hand side of the equations: the numbers not multiplying a variable- surely you do not mean the "solutions" to the equations!) depends on what you want to do with the matrix!

One way of solving a matrix equation, or system of equations converted to matrix form, is to find the inverse of the matrix multiplying the "unknown" vector and multiply both sides of the equation by it. In that case the matrix must include only the coefficients.

But I suspect you are talking about using row reduction. In that case, in order to apply the same row operations to the "right-hand side", you form the "augmented matrix", including the right-hand side as an additional column.
 
  • #3
HallsofIvy said:
It means that the rank of your matrix is one less than the dimension of the matrix. It also means that a system of equations having that matrix as coefficient matrix will not have a unique solution- either no solution or an infinite number of solutions.

Whether you include the "solutions of the equations" (by which I think you mean the right-hand side of the equations: the numbers not multiplying a variable- surely you do not mean the "solutions" to the equations!) depends on what you want to do with the matrix!

One way of solving a matrix equation, or system of equations converted to matrix form, is to find the inverse of the matrix multiplying the "unknown" vector and multiply both sides of the equation by it. In that case the matrix must include only the coefficients.

But I suspect you are talking about using row reduction. In that case, in order to apply the same row operations to the "right-hand side", you form the "augmented matrix", including the right-hand side as an additional column.

exactly what I was asking. Thanks a bunch.
 

What is row echelon form?

Row echelon form is a way of organizing a matrix where all of the non-zero rows are above any rows of all zeroes, and the leading coefficient (the first non-zero entry) of each row is to the right of the leading coefficient of the row above it.

Why is row echelon form important?

Row echelon form is important because it allows for easier computation of matrix operations such as finding the inverse or solving systems of equations. It also provides a standardized way of representing matrices, making it easier to compare and analyze them.

How do you put a matrix into row echelon form?

To put a matrix into row echelon form, you must use row operations such as swapping rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another. The goal is to get all of the non-zero entries in the matrix to line up on the left side of the matrix, with each subsequent row having more leading zeroes than the row above it.

Can all matrices be put into row echelon form?

No, not all matrices can be put into row echelon form. For example, a matrix with all zero entries cannot be put into row echelon form because there is no leading coefficient to the left of the leading coefficient of the row above it. Additionally, matrices with infinitely many solutions or no solutions cannot be put into row echelon form.

What are the limitations of row echelon form?

Row echelon form does not take into account the actual values of the entries in a matrix, only their positions. This means that different matrices can have the same row echelon form. Additionally, row echelon form does not apply to non-square matrices, as they cannot be inverted or used to solve systems of equations.

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