A square matrix A (non-singular) of order n is said to be invertible, if there exists a square matrix B of the same order such that

`AB= I_n = BA `


then B is called the inverse (reciprocal) of A and is denoted by `A^(-1) .`
. Thus,

`A^(-1) = B <=> AB= I_n= BA `

We have, `A ( adj A)= |A| I_n`

`A^(-1) A( adj A)= A^(-1) I_n |A| `

` I_n (adjA) = A^(-1) |A| I_n`

`A^(-1) =1/(detA) (text(adjoint of A)) `

or `A^(-1) = 1/(detA)(text(cofactor matrix of A)^T)`

`text(Note :)`

The necessary and sufficient condition for a square matrix A to be invertible is that `|A| != 0.`