Name origin: likely because it "determines" if a matrix is invertible or not, as a matrix is invertible iff determinant is not zero.

Matrix inverse ( $M^{- 1}$ )

When it exists, which is not for all matrices, only invertible matrix, the inverse is denoted:

M^{- 1}

Invertible matrix

The set of all invertible matrices forms a group: the general linear group with matrix multiplication. Non-invertible matrices don't form a group due to the lack of inverse.

When it distributes it inverts the order of the matrix multiplication:

(M N)^{T} = N^{T} M^{T}

(M^{T}) - 1 = (M^{- 1})^{T}

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