Inverse Matrix - Yousef's Notes
Inverse Matrix

Inverse Matrix

The inverse matrix of a square matrix $A$ is a matrix $A^{-1}$ that, when multiplied by $A$, produces the identity matrix $I$: $$ AA^{-1} = A^{-1}A = I $$ The inverse matrix exists only if the determinant of $A$ is non-zero: $$ \det(A) \neq 0 $$

The inverse matrix can be used to:

  • Solve systems of linear equations: $A\mathbf{x} = \mathbf{b}$ can be solved by multiplying both sides by $A^{-1}$: $\mathbf{x} = A^{-1}\mathbf{b}$
  • Find the inverse transformation: If $A$ represents a linear transformation, $A^{-1}$ represents the inverse transformation. For a $2 \times 2$ matrix:
$$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ The inverse matrix is: $$ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$

Note that the inverse matrix is unique, meaning that if $A$ has an inverse, it is the only matrix that satisfies the equation $AA^{-1} = I$.

#General Formula

$$ A^{-1} = \frac{1}{\det A} C^T $$ $$ \mathbf{x} = A^{-1} \mathbf{b} $$ $$ \mathbf{x} = \frac{1}{\det A} C^T \mathbf{b} $$

#Example

#Properties

$$ (A^T)^{-1} = (A^{-1})^T $$ $$ (AB)^{-1} = B^{-1}A^{-1} $$ $$ (A^{-1})^{-1} = A. $$