Solve 3x3 Matrix Equation: x, y, z Variables

[Destroxia]

Homework Statement

Find a 3x3 matrix A that satisfies the following equation where x, y, and z can be any numbers.

$A \begin{vmatrix} x \\ y \\ z \end{vmatrix} = \begin{vmatrix} x + y \\ x - y \\ 0 \end{vmatrix}$

Homework Equations

The Attempt at a Solution

I attempted to solve this like we learned in class, with gaussian elimination, but it obviously doesn't work in this scenario, because all of the coefficients in the matrix are literal in this sense.

I found the correct matrix intuitively, but I want to know how to do it properly for future, more complex problems, I've looked at the book, but couldn't find any sample problems regarding this type of solution.$A = \begin{vmatrix} a & b & c\\ d & e & f \\ g & h & i \end{vmatrix}$ <-- literal coefficients, not sure how to continue.

 

[Panphobia]

Think about how the matrix multiplication works, then you realize that the $A$ matrix is just a collection of coefficients infront of $x,y,z$ So in this case $$A =
\begin{bmatrix}
1 & 1 & 0 \\
1 & -1 & 0 \\
0 & 0 & 0
\end{bmatrix}
$$ To understand this yourself, try to generalize the entries of $A$ and then do the matrix multiplication with that column vector, and just equate coefficients.

 

[Destroxia]

Think about how the matrix multiplication works, then you realize that the $A$ matrix is just a collection of coefficients infront of $x,y,z$ So in this case $$A =
\begin{bmatrix}
1 & 1 & 0 \\
1 & -1 & 0 \\
0 & 0 & 0
\end{bmatrix}
$$ To understand this yourself, try to generalize the entries of $A$ and then do the matrix multiplication with that column vector, and just equate coefficients.

Okay, so it's just equating coefficients? Thank you very much!

 

[Panphobia]

Well to understand it you can think of it like that. For example in your case
$$
A
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}=
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix} =
\begin{bmatrix}
a_{11}x + a_{12}y + a_{13}z \\
a_{21}x + a_{22}y + a_{23}z \\
a_{31}x + a_{32}y + a_{33}z
\end{bmatrix} =
\begin{bmatrix}
x + y \\
x-y \\
0
\end{bmatrix}

$$

 

[ehild]

See the result on the three base vectors
$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$,

$\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$,

$\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$.