Rank of a Matrix - Explained by an Expert

In summary, the rank of a matrix is determined by the number of independent rows or columns. The row rank of a matrix is the number of linearly independent rows, and similarly, the column rank is the number of linearly independent columns.
  • #1
NonameNoface
6
0
Well this is eating my head ! or am I plain stupid ? ...

The rank of a Matrix , is determined by the number of independent rows or columns ..Fine .. here's a matrix ..

A = 2 4 1 3
-1 -2 1 0
0 0 2 2
3 6 2 5

Apparently " the second column is twice the first column, and that the fourth column equals the sum of the first and the third. The first and the third columns are linearly independent, so the rank of A is two "

here is the part where I am getting confused .. what if I consider the rows ..I see that all rows are linearly independent ... If somebody coudl explain this to me please ...

Thanks a lot
Regards
Mano
 
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  • #2
don't matter about those columns or rows. i think that you should only try to transorm that matrix into an echelon one and then you'll see yourself clearly independant rows which define the rank of that matrix
 
  • #3
The row rank of a matrix is the number of linearly independent rows, and similarly, the column rank is the number of linearly independent columns. Most importantly, note that the two are always equal!

With
[tex]
A = \left [ \begin{array}{cccc}
2 & 4 & 1 & 3 \\
-1 & -2 & 1 & 0 \\
0 & 0 & 2 & 2 \\
3 & 6 & 2 & 5
\end{array} \right ]
[/tex],
and since we know the column rank of this matrix is 2, can you now find a way to express any 2 of the rows as a linear combination of the others?
 
Last edited:
  • #4
NonameNoface said:
I see that all rows are linearly independent

Write down, don't just look.
 
  • #5
NonameNoface said:
Well this is eating my head ! or am I plain stupid ? ...

The rank of a Matrix , is determined by the number of independent rows or columns ..Fine .. here's a matrix ..

A = 2 4 1 3
-1 -2 1 0
0 0 2 2
3 6 2 5

Apparently " the second column is twice the first column, and that the fourth column equals the sum of the first and the third. The first and the third columns are linearly independent, so the rank of A is two "

here is the part where I am getting confused .. what if I consider the rows ..I see that all rows are linearly independent ... If somebody coudl explain this to me please ...

Thanks a lot
Regards
Mano
hey ... the rank of the matrix is 3.do the following operations to get echelon form...
r1<->r2;
r1<->-r1;
r2-->r2-2r1;
r4-->3r1-r4;
r4-->r3+r2+r4;
 
Last edited:
  • #6
madhusudan said:
hey ... the rank of the matrix is 2.do the following operations to get echelon form...
r1<->r2;
r1<->-r1;
r2-->r2-2r1;
r4-->3r1-r4;
r4-->r3+r2+r4;

can you get it
 
  • #7
NonameNoface said:
I see that all rows are linearly independent

Then perhaps a trip to the mathematical optician is in order. I can see that if I add the first row and twice the second row that...
 

1. What is the rank of a matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it is the dimension of the vector space spanned by the rows or columns of the matrix.

2. Why is the rank of a matrix important?

The rank of a matrix is important because it provides important information about the matrix, such as the number of solutions to a system of linear equations represented by the matrix. It also helps in determining the invertibility of a matrix and can be used for data compression and optimization problems.

3. How is the rank of a matrix calculated?

The rank of a matrix can be calculated using various methods, such as Gaussian elimination, singular value decomposition, or eigenvalue decomposition. These methods involve performing mathematical operations on the matrix to reduce it to a simpler form, from which the rank can be determined.

4. Can the rank of a matrix change?

Yes, the rank of a matrix can change depending on the operations performed on the matrix. For example, if two rows or columns of a matrix are multiplied by a constant, the rank remains the same. However, if two rows or columns are added together or if a row or column is multiplied by a non-zero constant and added to another row or column, the rank can change.

5. How is the rank of a matrix related to its determinant?

The rank of a matrix is related to its determinant in that a square matrix of rank n has a non-zero determinant if and only if its rank is n. This means that a matrix with a non-zero determinant has full rank, and a matrix with a zero determinant has reduced rank.

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