For this question, I know that [1, 1] * [?, ?] = [-1. 3] and [1. -1]*[?,?] = [-3,7] but I am confused how to go about finding a corresponding matrix that works for both of them.
Let the matrix be a 2x2 matrix be A with elements a, b, c, d.
And write the column matrices (vectors) as given (not as rows as you have).
Multiply A (on the left) by the each of the input vectors and equate with the image vectors.
This will give you two pairs of simultaneous equations. One pair will involve a and b, the other pair will involve c and d. Solve in the usual way.
Let us know if that works for you.
Here's a similar, but different, approach for you to try. Call your unknown matrix $A$. You are given$$
A \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}\text{ and }
A \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} -3 \\ 7 \end{bmatrix}
$$Expressing these two equations as a single matrix equation gives$$
A \begin{bmatrix} 1 & 1 \\1 & -1 \end{bmatrix} = \begin{bmatrix}-1 & -3 \\ 3 & 7 \end{bmatrix}$$Now can you solve for the matrix $A$?