Rank of Two Matrices: Is [B AB ... An-1B] = [AB A2B ... AnB]?

[WolfOfTheSteps]

Homework Statement

A is nxn
B is nxp

Is the rank of

[B AB ... A^n-1B]

equal to the rank of

[AB A²B ... AⁿB]?

If not, under what condition will the ranks be equal?

The Attempt at a Solution

I don't even know where to start. I know that both matrices should have the same number of pivots, but since this is so generalized I don't know how to learn anything about how multiplying by A would change the number of pivots!

Can someone help me out and at least give me a hint? I've been staring at this for the longest time and just don't know what to do!

Thanks!

 

[mighty2000]

my first approach would be to write out the matrices and try to find a pattern or relationship between the two. I would also look into the properties of matrix multiplication and how it affects the rank of a matrix.

One possible approach could be to consider the dimensions of the resulting matrices. Since A is nxn and B is nxp, the resulting matrices in both cases will have dimensions of nxp. This means that both matrices have the same number of columns, but the number of rows will depend on the number of matrices being multiplied.

For the first matrix [B AB ... An-1B], there will be n matrices being multiplied, so the resulting matrix will have n rows. For the second matrix [AB A2B ... AnB], there will also be n matrices being multiplied, so the resulting matrix will also have n rows.

Now, we know that the rank of a matrix is equal to the number of linearly independent rows (or columns) in the matrix. So, if we can show that the rows in both resulting matrices are linearly independent, then the ranks of the matrices will be equal.

Another approach could be to consider the null spaces of the matrices. The null space of a matrix is the set of all vectors that when multiplied by the matrix, result in a zero vector. If we can show that the null spaces of both matrices are equal, then it would mean that the ranks of the matrices are also equal.

Overall, this problem requires some knowledge of linear algebra and matrix operations. It might be helpful to review these concepts and try to apply them to this specific problem. I hope this helps!