Why must the eigenvalues be non-zero for the inverse matrix to exist?

The inverse matrix exists only for non-singular matrices, that is, those with a non-zero determinant. The eigenvalues of a matrix are related to its determinant: the determinant is equal to the product of all eigenvalues.

If at least one eigenvalue is zero, then the determinant will be zero, and the matrix will not be invertible.

Intuitively:

• Eigenvalues indicate how the matrix stretches or compresses space along certain directions.
• A zero eigenvalue means the matrix compresses space along some direction to zero, losing information and becoming non-invertible.

Thus, non-zero eigenvalues guarantee that the matrix preserves its dimensionality and has an inverse.