The Identity Matrix and Matrix Inverses
In scalar math, we have $5 \times 1 = 5$ and $5 \div 5 = 1$. To perform similar 'neutral' and 'inverse' logic with grids of numbers, we use the Identity Matrix and the Matrix Inverse.
The Identity Matrix ($I$)
The matrix equivalent of '1'. It is a square matrix with ones on the diagonal and zeros elsewhere.
Neutral Transformation
Multiplying any matrix $A$ by $I$ leaves $A$ unchanged ($AI = IA = A$). In geometry, this represents a transformation that does absolutely nothing to the space.
The Inverse ($A^{-1}$)
The matrix that 'undoes' an operation. If you apply transformation $A$ and then $A^{-1}$, you end up back at the start.
Existence and Singularity
Not all matrices have inverses. If a matrix is 'singular' or has zero determinant, it is irreversible. This often means that the data is redundant or missing critical dimensions required to map back to the origin.