# Inverse of a Matrix

For a square matrix $A$, its inverse, denoted as ${A}^{-1}$, is another matrix such that when multiplied with $A$, it results in the identity matrix $I$: $A×{A}^{-1}={A}^{-1}×A=I$

• Conditions:
• The matrix $A$ must be a square matrix.
• If the determinant of $A$ ($\text{det}\left(A\right)$ is non-zero, then $A$ has an inverse.
• If $\text{det}\left(A\right)=0$, the matrix $A$ is singular and does not have an inverse.

### Calculation of Inverse:

The inverse of a square matrix $A$ is calculated using the formula: ${A}^{-1}=\frac{1}{\text{det}\left(A\right)}\text{adj}\left(A\right)$

Where:

1. $\text{det}\left(A\right)$ is the determinant of matrix $A$.
2. $\text{adj}\left(A\right)$ is the adjoint (or adjugate) of matrix $A$.
3. For $2×2$ Matrix: For a matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the inverse ${A}^{-1}$ is given by: ${A}^{-1}=\frac{1}{\text{det}\left(A\right)}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ where $\text{det}\left(A\right)=ad-bc$ must not be zero.

### Example:

Consider a matrix $A$:

$A=\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]$
1. Determinant: $\text{det}\left(A\right)=\left(2×4\right)-\left(3×5\right)=8-15=-7$
2. Adjoint: From the previous example, the adjoint of $A$ is:
$\text{adj}\left(A\right)=\left[\begin{array}{cc}4& -3\\ -5& 2\end{array}\right]$
1. Inverse Calculation:
${A}^{-1}=\frac{1}{\text{det}\left(A\right)}\text{adj}\left(A\right)=\frac{1}{-7}\left[\begin{array}{cc}4& -3\\ -5& 2\end{array}\right]=\left[\begin{array}{cc}-\frac{4}{7}& \frac{3}{7}\\ \frac{5}{7}& -\frac{2}{7}\end{array}\right]$

4. For $3×3$ or Larger Matrices: The inverse of larger matrices can be calculated using various methods such as Gauss-Jordan elimination, cofactor expansion, or using software tools.

### Properties of the Inverse:

1. Inverse of Inverse: $\left({A}^{-1}{\right)}^{-1}=A$

2. Product with Inverse: $A×{A}^{-1}={A}^{-1}×A=I$

3. Transpose of Inverse: $\left({A}^{-1}{\right)}^{T}=\left({A}^{T}{\right)}^{-1}$

### Applications:

• Solving Systems of Equations: The inverse of a matrix is used to solve systems of linear equations in matrix form ($AX=B⇒X={A}^{-1}B$).

• Transformations: Inverse matrices are crucial in transforming vectors and shapes in geometry and computer graphics.