Given $T(x_1,x_2)$, find a matrix $A$ such that $A[x_1,x_2] = [T]$

Question

Suppose that $T(x_1,x_2) = (x_1+4x_2,0,x_1-3x_2,x_1)$. Find a matrix $A$ such that

$$A\begin{pmatrix} x_1\\x_2\end{pmatrix}=\begin{pmatrix} x_1 + 4x_2\\0\\x_1 - 3x_2\\x_1\end{pmatrix}$$

How would I solve this question?

Answer 1

Think about what it is you want your matrix $A$ to do. I.e., given a matrix $X = \begin{pmatrix} x \\ y \\ \end{pmatrix}$, you want to find a matrix $A$ such that $AX$ yields some other matrix whose rows are:

1. Once the first row of $X$ + 4 times the second row of $X$.
2. Zero
3. Once the first row of $X$ - 3 times the second row of $X$.
4. Once the first row of $X$.

Knowing the rules of matrix multiplication, can you take it from here?