Invertible matrix

In linear algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it can be multiplied by another matrix to yield the identity matrix. Invertible matrices are the same size as their inverse.

The inverse of a matrix represents the inverse operation, meaning if a matrix is applied to a particular vector, followed by applying the matrix's inverse, the result is the original vector.

An n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that$${\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},}$$where I_n denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the inverse of A, denoted by A⁻¹. Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix.

Consider the following 2-by-2 matrix:

The matrix ${\displaystyle \mathbf {A} }$ is invertible, as it has inverse ${\displaystyle \mathbf {B} ={\begin{pmatrix}2&3\\2&2\end{pmatrix}},}$ which can be confirmed by computing

${\displaystyle \mathbf {A} \mathbf {B} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}{\begin{pmatrix}2&3\\2&2\end{pmatrix}}={\begin{pmatrix}(-1)\times 2+{\tfrac {3}{2}}\times 2&(-1)\times 3+{\tfrac {3}{2}}\times 2\\1\times 2+(-1)\times 2&1\times 3+(-1)\times 2\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=\mathbf {I} _{2}}$

To check that it is invertible without finding an inverse, ${\textstyle \det \mathbf {A} =-{\frac {1}{2}}}$ can be computed, which is non-zero.

On the other hand, this is a non-invertible matrix: