What does it really mean to inverse a matrix ?

As we saw earlier , multiplying a matrix means doing a transformation . The inverse matrix will do just the opposite transformation .

Invertible / Nonsingular matrix

• An n x n $A$ matrix is said to be invertible if there is an n x n matrix $C$ such that $CA = I$ and $AC = I$

• A matrix is invertible if the determinant is not 0 (more on this later)

• A matrix is invertible if for each $b$ in $R^n$ , the equation $Ax = b$ has the unique solution $x = A^{-1}b$ . The real meaning of this sentence can be understood if you have a good understanding of matrix transformations some note earlier .

  You will see that the matrix transformation puts every vector to a new place

  Properties