Suppose I have
$X_1 = aY_1+bY_2$
$X_2 = cY_1+dY_2$
How is the quickest (or most efficient way ) to find the inverse functions?
The current way I am doing it is attempting to solve for Y1 in the first equation and plug it into the second equation.
But there is there some kind of a matrix trick or a fast way to solve these kind of problems?
Here is the matrix
$$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$
You should first invert this matrix. Assume the inverse of this matrix is
$$\begin{bmatrix}r&t\\g&h\end{bmatrix}$$
Then you will write
$$y_1=rx_1+tx_2$$ and $$y_2=gx_1+hx_2$$