1. The Concept: What Is an Inverse Matrix?
Definition
Given a square matrix $A$ of size $n \times n$, the inverse of $A$, written $A^{-1}$, is the unique matrix satisfying:
$$A\,A^{-1} = A^{-1}A = I$$where $I$ is the $n \times n$ identity matrix. In words, $A^{-1}$ undoes the effect of multiplying by $A$.
Analogy with Reciprocals
For a nonzero number $a$, the reciprocal $a^{-1} = \tfrac{1}{a}$ satisfies $a \cdot a^{-1} = 1$. The inverse matrix plays exactly the same role, with the identity matrix $I$ standing in for the number $1$.
Uniqueness (Three-Line Proof)
Suppose both $B$ and $C$ are inverses of $A$. Then:
$$B = BI = B(AC) = (BA)C = IC = C$$So $B = C$. The inverse, when it exists, is unique.
2. The Criterion: When Does the Inverse Exist?
Invertibility Theorem
A square matrix $A$ is invertible if and only if $\det(A) \neq 0$.
Vocabulary
| Term | Meaning |
|---|---|
| Nonsingular (invertible) | $\det(A) \neq 0$; $A^{-1}$ exists |
| Singular (non-invertible) | $\det(A) = 0$; $A^{-1}$ does not exist |
Why the determinant controls invertibility
Direction 1 ("if invertible, then $\det \neq 0$"): Suppose $A^{-1}$ exists. By the multiplicativity of determinants:
$$\det(A)\,\det(A^{-1}) = \det(A\,A^{-1}) = \det(I) = 1$$So $\det(A)$ cannot be zero, and moreover:
$$\det(A^{-1}) = \frac{1}{\det(A)}$$Direction 2 ("if $\det \neq 0$, then invertible"): The adjugate construction below explicitly builds $A^{-1}$ whenever $\det(A) \neq 0$, supplying the converse.