Rank of 5x3 matrix A is 3, what is RREF(A)?

In summary, the rank of a matrix is the number of linearly independent rows or columns it contains, a 5x3 matrix has 5 rows and 3 columns, a matrix with a rank of 3 means that there are 3 linearly independent rows or columns, RREF(A) stands for Row Reduced Echelon Form and is calculated using row operations.
  • #1
pyroknife
613
3

Homework Statement


Matrix A is of size 5x3 (5 rows and 3 columns) with rank(A)=3. Find the reduced row echlon form of A

The Attempt at a Solution


Rank(A)=3 thus, there are 3 pivot variables. Since there are 3 pivot variables and 3 columns=> no free variables, thus we have 2 rows of zeroes at the bottom. The top 3 rows represent a 3x3 identity matrix.
[/B]
It seems like the answer is just
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
Could someone please verify?I was curious if the question was instead a 3x5 matrix.
For this scenario I observe the following:
1) There will be 3 pivots, but since there are 5 columns, there will be 2 free variables.
2) I think RREF(A) can be 4!=4*3*2=24 different matrices?
 
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  • #2
pyroknife said:

Homework Statement


Matrix A is of size 5x3 (5 rows and 3 columns) with rank(A)=3. Find the reduced row echlon form of A

The Attempt at a Solution


Rank(A)=3 thus, there are 3 pivot variables. Since there are 3 pivot variables and 3 columns=> no free variables, thus we have 2 rows of zeroes at the bottom. The top 3 rows represent a 3x3 identity matrix.
[/B]
It seems like the answer is just
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
Could someone please verify?
Sure, this is fine.
pyroknife said:
I was curious if the question was instead a 3x5 matrix.
For this scenario I observe the following:
1) There will be 3 pivots, but since there are 5 columns, there will be 2 free variables.
2) I think RREF(A) can be 4!=4*3*2=24 different matrices?
And the rank is still 3?
Actually, there will be an infinite number of matrices. There would be two columns that could have any values.
 

What is the rank of a matrix?

The rank of a matrix is the number of linearly independent rows or columns it contains. It is also equal to the maximum number of linearly independent rows or columns that can be chosen from the matrix.

What is a 5x3 matrix?

A 5x3 matrix is a matrix with 5 rows and 3 columns. It can also be referred to as a 5 by 3 matrix.

What does it mean for a matrix to have a rank of 3?

A matrix with a rank of 3 means that there are 3 linearly independent rows or columns in the matrix. This also means that the matrix cannot be reduced any further without losing any of its linearly independent rows or columns.

What is RREF(A)?

RREF stands for Row Reduced Echelon Form. It is a way of representing a matrix in its most simplified form, where all the rows and columns are linearly independent and the leading entry of each row is a 1. It is useful for solving systems of linear equations and finding the rank of a matrix.

How is RREF(A) calculated?

RREF(A) is calculated using row operations, such as swapping rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another. These operations do not change the solutions to a system of linear equations, but they do simplify the matrix to its reduced form.

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