Summary On Inverse of a square Matrix and Applications
- A square matrix A with order n is said to be invertible or non- singular if and only if there is a square matrix B such that$AB =I_{n} =BA$, where B is an identity matrix having the same order as A.
- The inverse of A is denoted by $A^{-1}$. And $A^{-1} \ne 0$
- A matrix that doesn’t have an inverse is singular
- you solve the linear system using the concept of mtrix invertiblity: $AX = B,$ where $A$ = the coefficient matrix and $B$ = the constant vector or column matrix while$X$ = the set of variables.
- let $AX = B$ $\implies$ $x = A^{-1}B$
- Let A and B be invertible matrices of the same order, then: $A^{-1}$ is invertible and $(A^{-1})^{-1} = A$ $A^{T}$ is invertible and $(A^{T})^{-1} = (A^{-1})^{T}$. AB is invertible and $(AB)^{-1} = B^{-1}A^{-1}$.