Introduction: Undoing Transformations
Just as dividing undoes multiplication, the inverse matrix undoes a matrix transformation. The identity matrix is the "do nothing" transformation—the matrix equivalent of multiplying by 1.
The Identity Matrix
The identity matrix \(I\) has 1s on the diagonal and 0s elsewhere:
\[I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\]For any matrix \(A\): \(AI = IA = A\)
The Inverse Matrix
If \(A\) is invertible, there exists a unique matrix \(A^{-1}\) such that:
\[AA^{-1} = A^{-1}A = I\]A matrix is invertible if and only if \(\det(A) \neq 0\).
2×2 Inverse Formula:
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\]Worked Examples
Example 1: Computing an Inverse
Find the inverse of \(A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\):
\(\det(A) = 2 \cdot 3 - 1 \cdot 5 = 1\)
\[A^{-1} = \frac{1}{1} \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}\]Example 2: Verification
\[AA^{-1} = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix} \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \checkmark\]Example 3: Non-Invertible Matrix
\(B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}\) has \(\det(B) = 0\).
No inverse exists—the transformation loses information (collapses 2D to 1D).
The Quantum Connection
In quantum mechanics, the identity operator \(\hat{I}\) leaves all states unchanged: \(\hat{I}|\psi\rangle = |\psi\rangle\). Unitary operators (time evolution) are always invertible: \(U^{-1} = U^\dagger\). This ensures that quantum evolution is reversible—information is never lost (until measurement collapses the state).