Invertible Matrices and Uniqueness of Inverse

Definition

Definition. A square matrix $A$ of order $n$ is said to be invertible (or non-singular) if there exists a square matrix $B$ of the same order $n$ such that

where $I$ is the identity matrix of order $n$ (Section 2). The matrix $B$ is called the inverse of $A$, written $A^{-1}$. Only square matrices can possibly be invertible, since $I$ itself must be square and $AB$, $BA$ must both equal it. A square matrix that is not invertible is called singular.

Uniqueness of the Inverse

Theorem. If a square matrix $A$ has an inverse, that inverse is unique -- no square matrix can have two different inverses.

Proof. Suppose $B$ and $C$ are both inverses of $A$, so that

Consider the product $B(AC)$. Since $AC=I$, this is simply $BI=B$. On the other hand, by the associativity of matrix multiplication (Section 5), $B(AC)=(BA)C$; and since $BA=I$, this equals $IC=C$. Both computations evaluate the same product $B(AC)$, so their results must be equal:

Hence $B=C$ -- the two supposed inverses are in fact the same matrix. $\blacksquare$ This is precisely why the notation $A^{-1}$ (a single, definite symbol) is justified: there is never any ambiguity about which matrix it refers to. (Exercise: Invertible Matrices, Q4 applies this theorem to a case where two differently-written candidate inverses turn out, on inspection, to be the same matrix all along -- exactly as the theorem guarantees must happen.)

Finding the Inverse of a $2\times2$ Matrix

A full determinant-and-adjoint theory belongs to a later chapter, but the following shortcut is enough to compute the inverse of any $2\times2$ matrix here. For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, define its determinant $\det A=ad-bc$. If $\det A\neq0$, then $A$ is invertible, and

-- obtained by swapping the two diagonal entries, negating the two off-diagonal entries (together, this rearranged matrix is called the adjoint of $A$), and dividing every entry by $\det A$. If $\det A=0$, no such $B$ can exist and $A$ is singular (Exercise: Invertible Matrices, Q3 gives a worked example of this case). Example 9 and Exercise: Invertible Matrices, Q1--Q2 apply this shortcut and then verify the result directly against the definition $AA^{-1}=A^{-1}A=I$. …