Inverse

The inverse of a matrix is like the “undo” button for linear transformations. When you multiply a matrix by its inverse, you get the identity matrix— which acts like the number 1 for matrices. 🔄

Inverse of a 2×2 Matrix

For a matrix
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

the inverse \(A^{-1}\) is defined as

\[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

where the determinant is

\[ \det(A) = ad - bc \]

\(A\) must be invertible, meaning \(\det(A) \neq 0\).

If \(\det(A) = 0\), the inverse does not exist (the matrix is singular).