Raising a matrix \(A\) to a power \(n\) involves multiplying the matrix by itself \(n\) times. The matrix must be square.

Positive Power (n > 0):

\(A^n = \underbrace{A \times A \times \dots \times A}_{n \text{ times}}\)

Zero Power (n = 0):

By definition, any square matrix raised to the power of 0 is the identity matrix \(I\).

\(A^0 = I\)

Negative Power (n < 0):

A matrix is raised to a negative power by first finding its inverse, then raising the inverse to the corresponding positive power.

\(A^{-n} = (A^{-1})^n\)

This is only possible if the matrix is invertible (i.e., its determinant is not zero).