Inverse of a 2x2 Matrix

For a $2\times 2$ matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the inverse is $\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$.

Compute the determinant first, then swap and negate as needed.

1. 1. Compute the determinant

   The determinant is $ad-bc=2(1)-1(1)=1$.

2. 2. Apply the 2x2 inverse formula

   Swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant.

   $$A^{-1}=\frac{1}{1}\left(\begin{array}{cc}1& -1\\ -1& 2\end{array}\right)=\left(\begin{array}{cc}1& -1\\ -1& 2\end{array}\right)$$

Answer: [[1,-1],[-1,2]]