Matrix Inverses

Undoing a Matrix

The inverse of a square matrix $A$ is the matrix $A^{-1}$ that undoes it: $AA^{-1} = A^{-1}A = I$, where $I$ is the identity.

A matrix is invertible (or nonsingular) if and only if $\det A \neq 0$. Singular matrices ($\det A = 0$) collapse space, destroying information that cannot be restored.

For a $2 \times 2$ matrix, use this formula: $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Swap diagonal entries, negate off-diagonal ones, and divide by the determinant. Division by $\det A$ shows why zero determinants have no inverse.

Solving and Pitfalls

The primary use of the inverse is solving linear systems. If $A\mathbf{x} = \mathbf{b}$ and $A$ is invertible, multiply by $A^{-1}$ to get: $$\mathbf{x} = A^{-1}\mathbf{b}$$

This gives the unique solution directly, though Gaussian elimination is faster for large systems.

Common Pitfalls to Avoid:

• Not every matrix is invertible. Only matrices with a nonzero determinant have an inverse.
• Do not split products incorrectly. The inverse of a product is $(AB)^{-1} = B^{-1}A^{-1}$, not $A^{-1}B^{-1}$.

Multiplication is non-commutative, so the undoing steps must happen in the exact reverse sequence.