Is AB Invertible If n < m and B has a Non-Trivial Kernel?

[Buffu]

If $A$ is $m \times n$ matrix, $B$ is an $n \times m$ matrix and $n < m$. Then show that $AB$ is not invertible.

Let $R$ be the reduced echelon form of $AB$ and let $AB$ be invertible.

$I = P(AB)$ where $P$ is some invertible matrix.

Since $n < m$ and $B$ is $n \times m$ therefore there is a non trivial solution to $B\mathbf X = 0$
Let it be $\bf X_0$

$I\mathbf X = P(AB)\mathbf X_0 \implies \mathbf X_0 = PA (B \mathbf X_0) = PA \times 0 = 0$

Which means $\mathbf X_0 = 0$, which is contradiction. Therefore $AB$ is not invertible.

Is this correct ?

 

[StoneTemplePython]

This look ok, though I don't really think the proof by contradiction is necessary?

Notation wise, I would probably have used $\mathbf x$ to denote a vector in $B \mathbf x = 0$ -- as is, I'm not sure if the X is a matrix or vector, but it would be customary to use a vector here. Further, I wouldn't write $A \times 0$... that looks like a cross product?

I typically avoid proofs by contradiction if feasible. It occurs to me that when you note that $B \mathbf x_0 = 0$ for some $\mathbf x_0 \neq 0$, this tells us right away that $AB$ is not invertible. By definition an invertible matrix does not map any non-zero vector to the zero vector and $(AB)\mathbf x_0 = A(B\mathbf x_0) = A \mathbf 0 = \mathbf 0$, but clearly $(AB)$ does so and hence is not invertible.

 

[Buffu]

This look ok, though I don't really think the proof by contradiction is necessary?

Notation wise, I would probably have used $\mathbf x$ to denote a vector in $B \mathbf x = 0$ -- as is, I'm not sure if the X is a matrix or vector, but it would be customary to use a vector here. Further, I wouldn't write $A \times 0$... that looks like a cross product?

I typically avoid proofs by contradiction if feasible. It occurs to me that when you note that $B \mathbf x_0 = 0$ for some $\mathbf x_0 \neq 0$, this tells us right away that $AB$ is not invertible. By definition an invertible matrix does not map any non-zero vector to the zero vector and $(AB)\mathbf x_0 = A(B\mathbf x_0) = A \mathbf 0 = \mathbf 0$, but clearly $(AB)$ does so and hence is not invertible.

Thank you, I made it more complicated :(. I used bold to indicate it is a vector.

Why do you avoid proof by contradiction ?

 

[StoneTemplePython]

I suppose it is a matter of taste, but if I can show something directly, I try to show it directly. It tends to make a lot more sense to me, I can generally directly attach it into a lattice work of other ideas and theorems, is easier to remember and so on.

The tradeoff is that some things can only be shown by contradiction, or in some other cases, perhaps the proof by contradiction is a paragraph and a direct proof (say, a derivation) takes 3 pages. So you have to pick your spots.

Perhaps more idiosyncratically, I would add that I've become fond of proofs that have a "visual" component to them. In this case, we could note

$AB = \mathbf a_1 \widetilde{\mathbf b_1}^T + \mathbf a_1 \widetilde{\mathbf b_2}^T + ... + \mathbf a_n \widetilde{\mathbf b_n}^T$

which is to say $(AB)$ is an m x m matrix that is comprised of a series of $n$ rank one updates. When you add $n$ rank one matrices, you have a matrix with at most rank $n$. Hence $(AB)$ has at most rank $n \lt m$ and isn't invertible.

 

[Buffu]

I suppose it is a matter of taste, but if I can show something directly, I try to show it directly. It tends to make a lot more sense to me, I can generally directly attach it into a lattice work of other ideas and theorems, is easier to remember and so on.

The tradeoff is that some things can only be shown by contradiction, or in some other cases, perhaps the proof by contradiction is a paragraph and a direct proof (say, a derivation) takes 3 pages. So you have to pick your spots.

Perhaps more idiosyncratically, I would add that I've become fond of proofs that have a "visual" component to them. In this case, we could note

$AB = \mathbf a_1 \widetilde{\mathbf b_1}^T + \mathbf a_1 \widetilde{\mathbf b_2}^T + ... + \mathbf a_n \widetilde{\mathbf b_n}^T$

which is to say $(AB)$ is an m x m matrix that is comprised of a series of $n$ rank one updates. When you add $n$ rank one matrices, you have a matrix with at most rank $n$. Hence $(AB)$ has at most rank $n \lt m$ and isn't invertible.

I also like that type of proofs but the more elegant a proof is the more difficult it is to derive.

 

[member 587159]

I also like that type of proofs but the more elegant a proof is the more difficult it is to derive.

Don't worry about it. Your proof worked fine.

 

[WWGD]

If $A$ is $m \times n$ matrix, $B$ is an $n \times m$ matrix and $n < m$. Then show that $AB$ is not invertible.

Let $R$ be the reduced echelon form of $AB$ and let $AB$ be invertible.

$I = P(AB)$ where $P$ is some invertible matrix.

Since $n < m$ and $B$ is $n \times m$ therefore there is a non trivial solution to $B\mathbf X = 0$
Let it be $\bf X_0$

$I\mathbf X = P(AB)\mathbf X_0 \implies \mathbf X_0 = PA (B \mathbf X_0) = PA \times 0 = 0$

Which means $\mathbf X_0 = 0$, which is contradiction. Therefore $AB$ is not invertible.

Is this correct ?

EDIT OWise, you seem to be saying/arguing that the kernel of B is contained in the kernel of AB, which is true, and proves the point.
Just to nitpick a bit; you may want to put an index in your identity I , to know whether it is the $n \times n ; m \times m$, etc. Here , it is $n \times n$ , by matrix product rules. Specially if you are doing proofs, it is a good idea.