Matrix Inverses

Undoing a Matrix

The inverse of a square matrix $A$ is the matrix $A^{-1}$ that "undoes" it: $AA^{-1} = A^{-1}A = I$, the identity. Not every matrix has one — a matrix with an inverse is invertible (or nonsingular), and the criterion is exactly the determinant: $A$ is invertible if and only if $\det A \neq 0$ (Lesson 7). Singular matrices ($\det A = 0$) have no inverse because they collapse space, and collapsing loses information that no matrix can restore.

For a $2 \times 2$ matrix there is a tidy 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 the diagonal entries, negate the off-diagonal ones, and divide by the determinant. The division by $\det A$ makes the "no inverse when $\det = 0$" rule vivid — you cannot divide by zero.

The inverse's chief use is solving linear systems. If $A\mathbf{x} = \mathbf{b}$ and $A$ is invertible, multiply both sides by $A^{-1}$: $$\mathbf{x} = A^{-1}\mathbf{b}.$$ This gives the unique solution directly. (In practice, for large systems, Gaussian elimination is faster than computing the inverse, but conceptually the inverse packages the solution.)

Key algebraic facts: the inverse is unique when it exists; $(A^{-1})^{-1} = A$; and the inverse of a product reverses the order, $(AB)^{-1} = B^{-1}A^{-1}$ (like putting on socks then shoes, and reversing to take them off). This order-reversal mirrors the non-commutativity of matrix multiplication.

Common pitfall: thinking every square matrix has an inverse, or writing $(AB)^{-1} = A^{-1}B^{-1}$. Only matrices with nonzero determinant are invertible; singular matrices have none. And the inverse of a product reverses the order: $(AB)^{-1} = B^{-1}A^{-1}$, not $A^{-1}B^{-1}$ — because multiplication is non-commutative, the undoing must happen in the opposite sequence.