Computing a 2×2 Inverse
For a 2\times 2 matrix there is a tidy recipe for the
inverse.
Swap the two diagonal entries, negate the other two, and divide everything by the
determinant:
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \;\Rightarrow\; A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.
"Swap a and d, flip the sign of
b and c, divide by the determinant." If the
determinant is zero you'd be dividing by zero — which is the formula's way of telling you the
matrix has no inverse.
Build the inverse live
Set the entries of A and the inverse is computed for you by the recipe
above, alongside the determinant. Push the entries until the determinant hits zero and the
formula breaks — exactly as it should, because no inverse exists there.
A quick sanity check
You can always verify an inverse by multiplying:
A^{-1}A should come out to the identity
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. Bigger matrices use
Gaussian
elimination instead of a one-line formula, but the 2×2 recipe is worth memorizing —
it appears constantly, from solving little systems to the closed-form fit of
linear regression.