For $$A = \begin{bmatrix}a&-b\\ b&a\end{bmatrix}$$ find $a$ and $b$ such that $T(12,5) = (13,0)$.
I can't figure out how to set this problem up to solve it. I've tried row reducing an augmented matrix after I made a system of linear equations but that didn't work.
Update
I used the system of equations
a - b = 12
a + b = 5
and then row reduced the augmented matrix of that system. When I plugged those solutions into the original transformation matrix A and multiplied it by (12,5), it didn't give me the (13,0) result.
Hint 1: What is $T(12,5)$? Can you write down a formula for it, in terms of $a$ and $b$?
Hint 2: Now set $$T(12,5) = \left[\begin{array}{c}13\\0\end{array}\right].$$ This is a pair of equations with two unknowns. Can you solve it?