How many 3x3 matrices can you find

[rofln]

Homework Statement

My instructor wants to know if there are finite or infinite amount of solutions

Homework Equations

Matrix Multiplication

The Attempt at a Solution

I pretty much turned A into a 3x3 matrix like this...
| A₁₁ A₁₂ A₁₃ |
| A₂₁ A₂₂ A₂₃ |
| A₃₁ A₃₂ A₃₃ |

and then multiplied it by the given matrix of x y and z.

After multiplying I found myself with 2 matrices that I think would fit the solution...

| y 0 0 |
| 0 0 0 |
| 0 0 0 |

and

| 0 x 0 |
| 0 0 0 |
| 0 0 0 |

The first matrix would correspond to xy + 0y + 0z
0x + 0y + 0z
0x + 0y + 0z
yielding the answer indicated by the question


and the second would correspond to 0x + xy + 0z
0x + 0y + 0z
0x + 0y + 0z
yielding the answered indicated by the question

However, I find myself second guessing myself. I think I have 2 solutions, but since x and y are variables, they could be any number, ergo have infinite matrices. I think I may be over complicating things or perhaps I am just incorrect. Pretty much my question is, am I correct or perhaps on the right track or just wrong?

 

[DrClaude]

After multiplying I found myself with 2 matrices that I think would fit the solution...

| y 0 0 |
| 0 0 0 |
| 0 0 0 |

and

| 0 x 0 |
| 0 0 0 |
| 0 0 0 |

This indeed doesn't work. You have to find a matrix that will work with any vector, and return a vector with only the first component non-zero, such that this component is the product of the first two components of the original vector.

 

[haruspex]

Consider some specific choice(s) of x and y.

 

[rofln]

So working off of:

A₁₁x + A₁₂y + A₁₃z
A₂₁x + A₂₂y + A₂₃z
A₃₁x + A₃₂y + A₃₃z

I should only be concerned with the first row? And in that first row I must somehow make it so it equals the desired result of xy? I feel like I've hit a brick wall on this problem. I do not understand how get from addition to multiplication. I have tried substituting random values for x and y trying to factor them out, but I feel like I am just shooting in the dark with that. Am I supposed to try to find some factor values and work off of those? I apologize if I am not grasping this quite well, this matrix stuff is new to me.

 

[Dick]

So working off of:

A₁₁x + A₁₂y + A₁₃z
A₂₁x + A₂₂y + A₂₃z
A₃₁x + A₃₂y + A₃₃z

I should only be concerned with the first row? And in that first row I must somehow make it so it equals the desired result of xy? I feel like I've hit a brick wall on this problem. I do not understand how get from addition to multiplication. I have tried substituting random values for x and y trying to factor them out, but I feel like I am just shooting in the dark with that. Am I supposed to try to find some factor values and work off of those? I apologize if I am not grasping this quite well, this matrix stuff is new to me.

Yes, just think about the first row. The entries of the matrix A have to be definite numbers, not variables like x or y I would think. Have you considered that there may be no such matrix A? Or did that say that the matrix A is allowed to contain the variables x,y and z?