Existence question about matrix and ranks.

In summary, the conversation discusses the idea of proving or disproving the existence of matrices A1, A2, ..., As such that their ranks are all 1 and their sum is a matrix A with a rank of 10. The conversation also explores the possibility of using proof by contradiction and suggests using 10 by 10 matrices with specific properties as an example to test the idea. Ultimately, it is proven that such matrices do indeed exist and the rank of their sum is 10.
  • #1
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i need to prove diprove that there exist matrices A1,A2,...,As such that rank Ai=1 for every i=1,...,s and A=A1+A2+...+As with rankA=10.
my feeling this is not true, i thought trying to prove this by ad absrudum, let us assume that they exist, then the rows of Ai are scalar multiple of one row vector, now I am trying to show that if this is so then rankA cannot be equal to 10, but I am stuck on that, can someone advise me on this problem?
 
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  • #2
What about about 10 10 by 10 matrices A1 with 1 in the first row, first column 0 everywhere else, A2 with 1 in the second row, second column, 0 everywhere else, A3 with 1 in the third row, third column, 0 everywhere else, etc. What is the rank of each of thosef? What is the rank of their sum?
 
  • #3
nice example, yes it does work, and A=I_10.
 

FAQ: Existence question about matrix and ranks.

What is a matrix and how is it related to existence?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. In the context of existence, a matrix can represent a system of equations or relationships that describe the existence of certain entities or phenomena.

How does the rank of a matrix impact its existence?

The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In terms of existence, a matrix with a full rank has a unique solution, while a matrix with a lower rank may have multiple or no solutions. Therefore, the rank of a matrix is closely related to its existence.

Can a matrix exist without having a rank?

No, a matrix must have a rank in order to exist. If a matrix has no rank, it means that all of its rows or columns are linearly dependent, and therefore there is no unique solution to the equations or relationships represented by the matrix.

How is the concept of existence applied in the study of matrices and ranks?

The concept of existence is fundamental in understanding the properties and behavior of matrices and their ranks. It helps us determine the solvability of equations and systems represented by matrices, and also provides insights into the structure and relationships between different matrices.

Are there any real-life applications of the existence question about matrix and ranks?

Yes, the concept of existence in matrices and ranks has numerous real-life applications in fields such as engineering, economics, computer science, and physics. For example, in engineering, matrices are used to model systems and analyze their behavior, while in economics, they are used to represent input-output relationships in production. Understanding the existence of these matrices is crucial in making accurate predictions and decisions.

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