# The Maximum Rank of a Matrix B Given AB=0 and A is a Full Rank Matrix

• TickleTackleTock
In summary, the maximum rank of a matrix B is equal to the rank of matrix A when their product AB equals zero and A is a full rank matrix. This is due to the fact that the rank of a matrix represents the number of linearly independent rows or columns, and when AB=0, it means that all linear combinations of the rows or columns in B are also present in A. Therefore, the maximum rank of B can only be as high as the rank of A.
TickleTackleTock

## Homework Statement

Suppose that AB = 0, where A is a 3 x 7 full rank matrix and B is 7 x 53. What is the highest possible rank of matrix B.

## The Attempt at a Solution

Since each column of B is in the null space of A, the rank of B is at most 4.

I don't understand why it is 4.

What operation do I need to perform here to understand this? I am not a student, I am just trying to remember Linear Algebra.

Instead of these unwieldy matrices, let us consider the linear transformations they represent. For the sake of simplicity, let's denote the dimension of a vector space by an index. Then we have ##A\, : \,V_7 \longrightarrow V_3## and ##B\, : \, V_{53} \longrightarrow V_7\,.## So ##B## sends at least ##53-7=46## dimensions to zero anyway. Thus its rank must be between ##7## and ##0##. ##A## on the other hand sends exactly ##4## dimensions to zero, as it is of full rank. But both applied in a row: $$V_{53} \stackrel{B}{\longrightarrow} V_7 \stackrel{A}{\longrightarrow} V_3$$ sends all ##53## dimensions to zero. Now how many can ##B## leave from those ##7##, which will be left for ##A## to be sent to zero?

TickleTackleTock
So, if Matrix A were instead a 8 x 3 matrix, then B would have a rank of five? What I am seeing is that it is the subtraction of columns from rows in order to find the rank of B that maps A to 0?

TickleTackleTock said:
So, if Matrix A were instead a 8 x 3 matrix, then B would have a rank of five? What I am seeing is that it is the subtraction of columns from rows in order to find the rank of B that maps A to 0?
In this case you couldn't multiply the matrices in the first place. This means for the transformations, that we need to explain, what happens between the 8-dimensional image of ##B## and the 7-dimensional space, on which ##A## is defined.

You can translate it into row and column actions, too. But as we don't have a certain example, we can assume that the matrices are already in a form which is nice:
$$A=\begin{bmatrix}I_3 & | & 0_4 \end{bmatrix}\, , \,B=\begin{bmatrix} B_7^{\,'}&| &0_{46}\end{bmatrix}$$
with the ##(3\times 3)## identity matrix ##I_3## and any ##(7 \times 7)## matrix ##B_7^{\,'}##. This wouldn't change the result but is easier to handle. Now calculate ##AB =0## and see what does this mean for ##B_7^{\,'}## and its maximal rank.

Here are a couple of formulas which also might help occasionally: https://en.wikipedia.org/wiki/Rank_(linear_algebra)#Properties
In your example above I basically used https://en.wikipedia.org/wiki/Rank–nullity_theorem

TickleTackleTock
Sorry about that. I mean a a 2 x 7 matrix. That way the numbers would change a bit.

So, in this case(Your new example), the Rank(A) = 3, Rank(B) = unknown and the Rank(AB) = 0. Which is exactly the same as the previous example. I appreciate your help but I think that I am going to read through an introductory book again. It is apparent that I don't remember enough. Thank you!

A couple other approaches:

1.) Work through the Sylvester rank Inequality proof. Then apply it here.
2.) Less general, using gramm schmidt in reals, you can reason that all columns both B are orthogonal to the rows of A, and hence this implies what?

## 1. What is the definition of rank in linear algebra?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.

## 2. How do you find the rank of a matrix?

The rank of a matrix can be found by performing row operations on the matrix to reduce it to row-echelon form, and then counting the number of non-zero rows. The number of non-zero rows is equal to the rank of the matrix.

## 3. Can a matrix have a rank of zero?

Yes, a matrix can have a rank of zero if all of its elements are equal to zero. This means that all of the rows and columns are linearly dependent, and the matrix is essentially a scaled version of the zero vector.

## 4. How does the rank of a matrix affect its invertibility?

A matrix is invertible if and only if its rank is equal to the number of columns or rows. If the rank is less than the number of columns or rows, the matrix is not invertible.

## 5. What is the relationship between the rank of a matrix and its determinant?

The determinant of a matrix is equal to the product of its eigenvalues. The rank of a matrix is equal to the number of non-zero eigenvalues. Therefore, the determinant of a matrix is zero if and only if the rank is less than the number of columns or rows.

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