Rank of Product Of Matrices

In summary, the rank of a matrix is equal to the number of linearly independent rows, and for two matrices A and B, the rank of their product AB is less than or equal to the rank of A and B individually. With this understanding, we can determine that the rank of AB is equal to the rank of A, which is m. This is because the rank of B is equal to its number of linearly independent columns, which is also n. Therefore, the rank of AB is m.
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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?
 

1. What is the definition of rank of product of matrices?

The rank of product of matrices is the maximum number of linearly independent rows or columns in the resulting matrix after multiplying two matrices together.

2. How is the rank of product of matrices calculated?

The rank of product of matrices is calculated by taking the smaller rank of the two matrices being multiplied. So if matrix A has a rank of 3 and matrix B has a rank of 5, the rank of the product AB will be 3.

3. What is the significance of the rank of product of matrices in linear algebra?

The rank of product of matrices is important in linear algebra because it provides information about the linear independence of the resulting matrix. It is also used in solving systems of linear equations and determining whether a matrix is invertible.

4. Can the rank of product of matrices be greater than the rank of the individual matrices?

No, the rank of product of matrices cannot be greater than the rank of the individual matrices. The rank of the product is limited by the smaller rank of the two matrices being multiplied.

5. How does the rank of product of matrices affect the number of solutions in a system of linear equations?

The rank of product of matrices can tell us whether a system of linear equations has a unique solution, infinitely many solutions, or no solution. If the rank of the product is equal to the number of variables in the system, then there is a unique solution. If the rank is less than the number of variables, there are infinitely many solutions. And if the rank is equal to 0, there are no solutions.

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