Definition

Let \(A\) be an \(n\times n\) matrix and let \(k\) be a natural number. We define \[A^0 = I_n\] and \[A^k = AA^{k-1}.\]

From this definition, one sees that \(A^1 = AA^0 = AI_n = A\) and \(A^2 = AA^1 = AA\).

Example

\begin{eqnarray} \begin{bmatrix} 1 & -1\\ 0 & 2\end{bmatrix}^3 & = & \begin{bmatrix} 1 & -1\\ 0 & 2\end{bmatrix} \begin{bmatrix} 1 & -1\\ 0 & 2\end{bmatrix}^2 \\ & = & \begin{bmatrix} 1 & -1\\ 0 & 2\end{bmatrix} \left(\begin{bmatrix} 1 & -1\\ 0 & 2\end{bmatrix} \begin{bmatrix} 1 & -1\\ 0 & 2\end{bmatrix}\right) \\ & = & \begin{bmatrix} 1 & -1\\ 0 & 2\end{bmatrix} \begin{bmatrix} 1 & -3\\ 0 & 4\end{bmatrix} \\ & = & \begin{bmatrix} 1 & -7\\ 0 & 8\end{bmatrix} \end{eqnarray}

If \(A\) is invertible, then \(A^k\) is also invertible and \( \left(A^k\right)^{-1}\) is given by \(\left(A^{-1}\right)^k\). We define \( A^{-k}\) to be \( \left(A^k\right)^{-1}\).

Computing powers of matrices does not seem as easy as computing powers of numbers. However, there are cases when matrix powers are relatively easy to compute as you will see in some of the exercises.

Quick Quiz

Exercises

Compute each of the following matrix powers.

1. \(\begin{bmatrix} 1 & 2 \\ -1 & 1\end{bmatrix}^3\)  

   \(\begin{bmatrix} -5 & 2 \\ -1 & -5\end{bmatrix}\)

2. \(\begin{bmatrix} 2 & 0 \\ 0 & 3\end{bmatrix}^6\)  

   \(\begin{bmatrix} 64 & 0 \\ 0 & 729\end{bmatrix}\) or \(\begin{bmatrix} 2^6 & 0 \\ 0 & 3^6\end{bmatrix}\)

3. \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & i\end{bmatrix}^{-2}\)  

   \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & -1\end{bmatrix}\)