Matrix Inverse

The inverse of a matrix $A$ is a matrix $A^{-1}$ such that:

$$AA^{-1} = A^{-1}A = I$$

2×2 Inverse Formula

For $A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$:

$$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$$

The denominator $ad - bc$ is the determinant. If it’s zero, no inverse exists.

When Does $A^{-1}$ Exist?

$A^{-1}$ exists (A is invertible) if and only if:

• $\det(A) \neq 0$
• $A$ has $n$ pivots
• $Ax = 0$ has only the trivial solution
• The columns are linearly independent
• The column space is all of $\mathbb{R}^n$

Properties

• $(AB)^{-1} = B^{-1}A^{-1}$ (reverse order!)
• $(A^T)^{-1} = (A^{-1})^T$
• $(A^{-1})^{-1} = A$