Find Matrix A for 2x2 Homework Equations

[Luscinia]

Homework Statement

(All matrices are 2x2. Sorry for the awkward formatting)

Find Matrix A such that
(1 -1)^-1 (2 3)
(0 1 ) = (1 2) A^T

Homework Equations

A₁A₂...A_k=A_k^-1...A₂^-1A₁^-1

The Attempt at a Solution

(1 -1)^-1 (2 3)
(0 1 ) = (1 2) A^T

(1 -1)^-1 (2 3)^-1
(0 1 ) (1 2) = A^T

((2 3)(1 -1))^-1
((1 2)(0 1)) = A^T

(2 1)^-1
(1 1) =A^T

(2 1|1 0) R1-R2 (1 0|1 -1) (1 0|1 -1)
(1 1|0 1) = (1 1|0 1) R2-R1 = (0 1|-1 2)

A^T=(1 -1)
___(-1 2)
A=(1 -1)
__(-1 2)

The answer was supposed to be (2 -1)
___________________________ (-1 1)

 

[th4450]

(1 -1)^-1 (2 3)
(0 1 ) = (1 2) A^T

(1 -1)^-1 (2 3)^-1
(0 1 ) (1 2) = A^T

ABA^-1 ≠ B
So for the LHS, you should put $\left(^{2 3}_{1 2}\right)$^-1 on the left.