Given the matrix $A = \begin{bmatrix}-2&1\\0&-1\end{bmatrix}$, find $A^3$
I don't understand what $A^3$ is supposed to represent? What are they asking me to find exactly??
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$\begingroup$For square matrices, $\underbrace{A^n=A\cdot A \cdot A \cdots A}_\text{n time}$ where $n \in \mathbb N$
So: $A^3 = A\cdot A \cdot A = $ $\begin{bmatrix}-2&1\\0&-1\end{bmatrix}$$\begin{bmatrix}-2&1\\0&-1\end{bmatrix}$$\begin{bmatrix}-2&1\\0&-1\end{bmatrix}$
The product operation of matrices is associative. you can evaluate $A^3$ easily
$\endgroup$ 2 $\begingroup$Diagonalize the matrix, where the non-zero elements will be the matrix eigenvalues, then the same matrix raised to any power will have the same eigenvalues raised to the required power associated with the same eigenvectors of the original matrix, thus yielding your desired matrix.
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