# Thread: Finding two matrices that multiply to...

1. ## Finding two matrices that multiply to...

First of all, what is meant by a numerical matrix?
Is it a matrix whose ij entries are not 0?
Is it a matrix whose ij entries have at least 1 number?

What are 2 numerical, 3x3, matrices A, B that multiply to the 0 matrix and A and B are not zero matrices themselves.

I have a simple solution to this depending on the definition of a numerical matrix. If it is the first definition then I need an example.

Also, how do I find a 3x3 matrix B such that for every 3x3 matrix A, AB = 3B?
I don't think it's possible, what if A is a zero matrix?

2. Originally Posted by Noxide

What are 2 numerical, 3x3, matrices A, B that multiply to the 0 matrix and A and B are not zero matrices themselves.
If $\displaystyle A \times B = 0$ then $\displaystyle A$ or $\displaystyle B$ must also be zero.

3. Originally Posted by pickslides
If $\displaystyle A \times B = 0$ then $\displaystyle A$ or $\displaystyle B$ must also be zero.

I don't think this is true at all... maybe for invertible matrices?
For example, let x1 = x2 = x3 = (1, -2, 1) = rows of the matrix A
and let y2 = y3 = y4 = (2, 2, 2) = columns of the matrix B
AB = 0...

What if we just let 1 of the matrices be invertible?

4. Originally Posted by Noxide
First of all, what is meant by a numerical matrix?
Is it a matrix whose ij entries are not 0?
Is it a matrix whose ij entries have at least 1 number?
A numerical matrix is simply a matrix whose entries are numbers. There is no requirement that the any of the entries be non-zero.

What are 2 numerical, 3x3, matrices A, B that multiply to the 0 matrix and A and B are not zero matrices themselves.

I have a simple solution to this depending on the definition of a numerical matrix. If it is the first definition then I need an example.
How about $\displaystyle A= \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$ and $\displaystyle B= \begin{bmatrix}1 & 1 \\ -1 & -1\end{bmatrix}$

Also, how do I find a 3x3 matrix B such that for every 3x3 matrix A, AB = 3B?
I don't think it's possible, what if A is a zero matrix?
Yes, you are right.

5. Originally Posted by pickslides
If $\displaystyle A \times B = 0$ then $\displaystyle A$ or $\displaystyle B$ must also be zero.
Sorry, but that is completely false! We are talking about matrices here, not numbers. The ring of matrices has "zero divisors".