What does it mean for a matrix to have rank 0 ( zero) ?

In summary, a matrix with rank 0 means that the image of the linear map it represents is the vector space 0. This also means that the matrix itself is the matrix 0. A matrix with at least one non-zero element will have at least rank 1, as the rank is the dimension of the row space and a non-zero element will result in a row space with dimension at least one.
  • #1
sauravrt
15
0
What does it mean for a matrix to have rank 0 ( zero) ?
 
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  • #2
The rank of a matrix is the dimension of the image of the linear map it represent. Since the only vector space of dimension 0 is the vector space denoted 0 consisting of only one elements (namely, 0), to say that a matrix is of rank 0 is to say that the image of the linear map it represents is the vector space 0.

You can convince yourself that, in turn, this implies that the matrix is the matrix 0 (the matrix having 0 in all its entries...)
 
  • #3
Thanks quasar987.

So a matrix with atleast one non-zero element will have atleast rank 1 ?
 
  • #4
What do you think? And why?
 
  • #5
Yes, the rank of a matrix is the dimension of the row space.

If a matrix has a zero-dimensional row space, it consists of a single vector - the zero vector. The space consisting of the zero vector only has dimension zero.

If a vector had an entry besides 0, then that row would not be the zero vector. Then the row space would include inifinitely many vectors corresponding to all scalar multiples of that vector... and have dimension at least one.
 

1. What does it mean for a matrix to have rank 0 (zero)?

Having a rank of 0 means that the matrix has no linearly independent rows or columns. In other words, all the rows and columns in the matrix can be expressed as linear combinations of each other.

2. Can a matrix have a rank of 0 (zero) if it is not a zero matrix?

Yes, a matrix can have a rank of 0 even if it is not a zero matrix. This happens when the rows and columns are not linearly independent, meaning they can be reduced to all zeros through row operations.

3. What is the relationship between a matrix's rank and its dimensions?

The rank of a matrix cannot be greater than the number of rows or columns it has. If a matrix has a rank of 0, it means that it has no linearly independent rows or columns, and therefore, the number of rows and columns must be equal.

4. How does a matrix's rank affect its invertibility?

A matrix with a rank of 0 is not invertible, meaning it does not have an inverse matrix. This is because the columns of a non-invertible matrix are linearly dependent, and cannot form a basis for the vector space.

5. Can a matrix with a rank of 0 (zero) still be used for solving systems of equations?

No, a matrix with a rank of 0 cannot be used to solve systems of equations. This is because a matrix with rank 0 does not have a pivot in any column, meaning it cannot be reduced to row echelon form and therefore cannot be used to solve equations using Gaussian elimination.

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