## Precalculus (6th Edition) Blitzer

Suppose there is an $n\times n$ square matrix A. If there exists another $n\times n$ matrix ${{A}^{-1}}$ such that: \begin{align} & A{{A}^{-1}}={{I}_{n}} \\ & {{A}^{-1}}A={{I}_{n}} \end{align} Then, the square matrix ${{A}^{-1}}$ is said to be a multiplicative inverse of the square matrix A. The result of multiplication of matrix A and ${{A}^{-1}}$ is an $n\times n$ square matrix, which is an identity matrix ${{I}_{n}}$. Only square matrices can have multiplicative inverses. And also, it is not necessary that every square matrix will possess an inverse matrix. Non-square matrices cannot have a multiplicative inverse. This is because of the following reason: A non-square matrix is a matrix that has a different number of rows and columns. If A is a $m\times n$ matrix and B is a $n\times m$ matrix ( $n\ne m$ ), then the products AB and BA will be of different orders. This means two products AB and BA are not equal to the same multiplicative identity matrix.