The Inverse of a 2×2 Matrix

Intuition

If a matrix is a transformation (stretch, rotate, shear), its inverse is the transformation that undoes it. A matrix that squashes the plane onto a line destroys information and cannot be undone — it has no inverse.

Precise explanation

For $A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $\det A = ad-bc \ne 0$:

Recipe: swap the diagonal, negate the off-diagonal, divide by the determinant. Check: $A A^{-1} = A^{-1} A = I$.

$|\det A|$ is the factor by which $A$ scales areas; $\det A = 0$ means area collapses to zero.

Numerically, solving $Ax=b$ directly (MA-08) is usually preferred over forming $A^{-1}$, but the inverse is a useful concept and appears in closed-form formulas such as $w = (X^\top X)^{-1}X^\top y$.

Worked example

$A = \begin{pmatrix}4&7\\2&6\end{pmatrix}$, $\det = 24 - 14 = 10$, $A^{-1} = \frac{1}{10}\begin{pmatrix}6&-7\\-2&4\end{pmatrix} = \begin{pmatrix}0.6&-0.7\\-0.2&0.4\end{pmatrix}$.

Common mistakes

• Negating the diagonal instead of the off-diagonal.
• Forgetting to divide by the determinant.
• Comparing $AA^{-1}$ to $I$ with == — use a tolerance.

Why this matters

Inverses appear in the normal equations, Gaussian covariance matrices and change-of-basis arguments throughout ML theory.

References

• Mathematics for Machine Learning — Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong (Cambridge University Press) (S009)