Inverse Matrix
Definition
Let \(A\) be a square matrix of size \(n\times n\). We say that \(A\) is invertible if there exists a matrix \(A^{-1}\) such that \(AA^{-1}=I\) and \(A^{-1}A=I\) where \(I\) is the identity matrix of size \(n\times n\). The matrix \(A^{-1}\) is called the inverse matrix of \(A\).
Conceptual Interpretation
If a matrix represents a linear transformation \(T(x)=Ax\) then the inverse matrix represents the transformation that reverses that effect:
$$ x=A^{-1}(Ax) $$
Therefore, the inverse matrix “undoes” the action of \(A\).
A matrix \(A\) has an inverse if and only if \(\det(A) \neq 0\). When \(\det(A)=0\) the matrix is called singular and it does not have an inverse.
Formula Using the Adjugate Matrix
If \(A\) is invertible, then
$$ A^{-1}=\frac{1}{\det(A)}\,\operatorname{adj}(A) $$
where \( \operatorname{adj}(A) \) is the adjugate matrix.
Example (\(2 \times2\) Case)
If
$$ A= \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$
then
$$ A^{-1}= \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$
provided that
$$ ad-bc\neq0. $$
Important Properties
If \(A\) and \(B\) are invertible matrices, then the following properties hold:
$$ (AB)^{-1}=B^{-1}A^{-1} $$
$$ (A^{-1})^{-1}=A $$
$$ I^{-1}=I $$
$$ (A^T)^{-1}=(A^{-1})^T $$
Interactive Examples
In this applet you will find examples of how to compute the inverse matrix. Press the “New” button to generate a different example, move the level slider to increase or decrease the difficulty of the exercise, and use the arrow buttons to move forward or backward through the steps of the procedure for computing the inverse matrix.