where In is the n-by-n identity matrix
If this equation is true, X is the inverse of A, denoted by A-1. A is also the inverse of X.
A matrix is invertible if and only if it possesses an inverse.
Every invertible matrix has only one inverse.
For example, if AX = I and AY = I, then X = Y. So, X = Y = A-1.
To prove this, consider the case of X.A.Y.
The inverse may be computed from the adjugate matrix, which shows that a matrix is invertible if and only if its determinant is itself invertible: over a field such as the real or complex numbers, this is equivalent to specifying that the determinant does not equal zero.