Solve Matrix A: Homework Equations & Solution

[ver_mathstats]

Homework Statement

Solve for the Matrix A.
(A^T + 4I)^-1 = [-1 1, 2 1]

Homework Equations

The Attempt at a Solution

I am unsure of how exactly to do this.

Here is what I have done:

(A^-1)^T = 1/4I + [-1 1, 2 1]

Am I on track?

Thank you.

 

[fresh_42]

Doesn't look correct, a bit confused to be honest. Do what you always do, if you solve for $x$: strip everything around it until there will be $x=...$
E.g. if we have $4x = 8$ we divided by $4$, the opposite of $4 \,\cdot \,$; if we have $4+x=8$, we subtracted $4$, the opposite of $4\,+\,$.
Now if we have any function $f(x)= 8$, we had to apply $f^{-1}$ the opposite of $f$ to get $x=f^{-1}(f(x))=f^{-1}(8)$.

Your example works the same, only that your $x$ is a matrix $A$. You have $f(g(A)+4)=B$, where $f$ is inversion and $g$ is transposition. Now strip $f$, then $+4$ and at last $g$ by doing the opposite.

 

[ver_mathstats]

Doesn't look correct, a bit confused to be honest. Do what you always do, if you solve for $x$: strip everything around it until there will be $x=...$
E.g. if we have $4x = 8$ we divided by $4$, the opposite of $4 \cdot$; if we have $4+x=8$, we subtracted $4$, the opposite of $4+$.
Now if we have any function $f(x)= 8$, we had to apply $f^{-1}$ the opposite of $f$ to get $x=f^{-1}(f(x))=f^{-1}(8)$.

Your example works the same, only that our $x$ is a matrix $A$. You have $f((g(A)+4)=B$, where $f$ is inversion and $g$ is transposition. Now strip $f$, then $+4$ and at last $g$ by doing the opposite.

I solved it, I realized I was doing it completely wrong which confused me as a result. The answer is 1/3[-13 2, 1 -11]. Thank you.