Matrix inversion method

Matrix inversion is a method for finding the inverse of a matrix.
The inverse of a matrix can be used to solve systems of linear equations, find the coefficients of a polynomial, and perform other mathematical operations.
The inverse of a matrix can be found using the adjoint of the matrix and the determinant of the matrix.
A matrix is invertible if and only if its determinant is nonzero.
Matrix inversion can be computationally expensive for large matrices, and other methods such as LU decomposition or Gaussian elimination may be preferred.

Let A be a square matrix of order n and B be the identity matrix of order n, then the inverse of A is given by $$A^{-1} = \frac{1}{|A|} adj\ A$$
The system has no solution if and only if the reduced row echelon form of the augmented matrix has a row of the form: $$\begin{bmatrix}0 & 0 & \\cdots & 0 & b\end{bmatrix}$$ where $b$ is a nonzero constant.
The system has infinitely many solutions if and only if the reduced row echelon form of the augmented matrix has a row of the form: $$\begin{bmatrix}0 & 0 & \\cdots & 0 & 0\end{bmatrix}$$