What Is the Rank of the Product of Two Matrices?

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SUMMARY

The rank of the product of two matrices, specifically when matrix A is an m x n matrix with rank m and matrix B is an n x p matrix with rank n, is determined to be rank(AB) = rank(B) = n. This conclusion is based on the properties of matrix ranks, where rank(AB) is less than or equal to both rank(A) and rank(B). The discussion emphasizes the importance of understanding linear independence and the relationship between the rank of a matrix and the dimension of its image.

PREREQUISITES
  • Understanding of matrix rank and linear independence
  • Familiarity with matrix multiplication
  • Knowledge of the properties of linear transformations
  • Basic concepts of image and kernel in linear algebra
NEXT STEPS
  • Study the properties of matrix multiplication and its effects on rank
  • Learn about the dimension of the image and kernel of linear transformations
  • Explore the concept of linear independence in more depth
  • Investigate the implications of the Rank-Nullity Theorem
USEFUL FOR

Students studying linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of matrix ranks and their applications in linear transformations.

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Homework Statement



Let A be an m * n matrix with rank m and B be an n * p matrix with rank n. Determine the rank of AB. Justify your answer.





Homework Equations





The Attempt at a Solution





I don't really know where to start off, but I have some things that might help me. I know that the rank of a matrix is equal to the number of linearly independent rows in it, and I also know that if A and B are two matrices, then rank(AB) <= rank(A) and also rank(AB) <= rank(B).

I'm sure if I was given a push in the right direction I could solve this.
 
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What's the relation between the rank of a matrix and the dimension of the image?
 

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