State whether each of the following matrices could be raised to a power:
A=\begin{bmatrix} 5 & 0 \\ 3 & 4 \end{bmatrix}
A=\begin{bmatrix} 5 & 4 & 1 \\ 0 & -1 & -2 \end{bmatrix}
A=\begin{bmatrix} 0 & 4 \\ 3 & -2 \\ -5 & 5 \end{bmatrix}
A=\begin{bmatrix} -2 & -5 & -4 \\ 2 & -3 & 5 \\ 0 & -1 & 3 \end{bmatrix}
For each matrix, find A^2:
A=\begin{bmatrix} 0 & -1 \\ -2 & 1 \end{bmatrix}
A=\begin{bmatrix} 2 & 1 & 0 \\ -4 & 5 & -1 \\ 3 & -2 & -5 \end{bmatrix}
Consider A=\begin{bmatrix} -1 & 0 \\ 5 & -2 \end{bmatrix}.
Find A^{2}.
Use the fact that A^{3} = A \times A^{2} to find A^{3}.
Consider A=\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}.
Find A^{2}.
Use the fact that A^{3} = A \times A^{2} to find A^{3}.
Find A^{2} \times A.
Does A^{3} equal A^{2} \times A?
For each of the following square matrices, use your CAS calculator to find the indicated power, rounding each element to three decimal places where necessary:
A=\begin{bmatrix} 6 & -9 \\ 4 & 8 \end{bmatrix},\, A^{2}
A=\begin{bmatrix} -5 & 3 \\ 2 & -1 \end{bmatrix},\, A^{7}
A=\begin{bmatrix} 0.85 & 0.46 \\ 0.15 & 0.54 \end{bmatrix},\, A^{27}
A=\begin{bmatrix} -2 & 3 & 0 \\ -2 & 1 & 6 \\ 0 & 4 & 0 \end{bmatrix},\, A^{6}
A=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix},\, A^{16}
A=\begin{bmatrix} 0.1 & 0.3 & 0.6 \\ 0 & 0.5 & 0.4 \\ 0.9 & 0 & 0.2 \end{bmatrix},\, A^{28}
A=\begin{bmatrix} 1 & -2 & 3 & 4 \\ 2 & 1 & -7 & 8\\ 3 & 5 & 8 & -6 \\ -8 & -1 & 10 & 9 \end{bmatrix},\, A^3
A=\begin{bmatrix} 0 & 0.4 & 0.7 & -0.1 \\ 0.9 & 0.8 & -0.7 & 0.2\\ -0.1 & 0.5 & 0.5 & 0.1 \\ 0 & 0.2 & 0.2 & 0.6 \end{bmatrix},\, A^{15}
A=\begin{bmatrix} 7 & 1 & 0 & 1 & 2 \\ 2 & 3 & 0 & 4 & 2\\ 1 & 2 & 4 & 1 & 4 \\ 2 & 1 & 4 & 3 & 0 \\ 0 & 5 & 2 & 1 & 2 \end{bmatrix},\, A^5
A=\begin{bmatrix} 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \end{bmatrix},\, A^{12}
A=\begin{bmatrix} 0 & 0 & 0 & 0.1 & 0 \\ 0.4 & 0 & 0.2 & 0.8 & 0\\ 0 & 0.5 & 0 & 0 & 0.6 \\ 0.3 & 0.3 & 0.3 & 0 & 0 \\ 0.2 & 0 & 0.6 & 0 & 0.4 \end{bmatrix},\, A^{27}
For each of the matrices below:
Find A^{20}, giving each element correct to three decimal places.
Find A^{21}, giving each element correct to three decimal places.
Hence or otherwise, describe the behaviour of the elements of matrix A as it is raised to larger powers.
A=\begin{bmatrix} 0.4 & 0 & 0.2\\ 0.2 & 0.9 & 0.5 \\ 0.4 & 0.1 & 0.3 \end{bmatrix}
A=\begin{bmatrix} 0\enspace & 0.1\enspace & 0\enspace & 0.6\\ 0.5\enspace & 0\enspace & 0.8\enspace & 0 \\ 0\enspace & 0.9\enspace & 0\enspace & 0.4 \\ 0.5\enspace & 0\enspace & 0.2\enspace & 0 \end{bmatrix}
The map shows four towns and the paths connecting them:
Let matrix A below, represent all of the single-step paths between the towns:
Find A^{4}, the matrix that represents all possible four-step paths between the towns.
How many four-step paths can be taken from Greenville to Dunham?
Five friends are participating in a puzzle event. As part of the rules of the event, each person can only send messages to two other people. Let matrix A represent the possible communication pathways:
\begin{matrix} & \text{Receiver} \\ \text{Sender} & \begin{matrix} \\& \begin{matrix} \text{Aaron} & \text{Bill} & \text{Clara} & \text{Darrel} & \text{Eve} \end{matrix} \\ \begin{matrix} \text{Aaron} \\ \text{Bill} \\ \text{Clara} \\ \text{Darrel} \\ \text{Eve} \end{matrix} & \begin{bmatrix} \quad 0 \qquad & 1 \qquad & 1 \qquad & 0 \qquad & 0 \quad \\ \quad 0 \qquad & 0 \qquad & 1 \qquad & 1 \qquad & 0 \quad \\ \quad 0 \qquad & 0 \qquad & 0 \qquad & 1 \qquad & 1 \quad \\ \quad 1 \qquad & 0 \qquad & 0 \qquad & 0 \qquad & 1 \quad \\ \quad 1 \qquad & 1 \qquad & 0 \qquad & 0 \qquad & 0 \quad \end{bmatrix} \end{matrix} \end{matrix}
Find A^{2}, the matrix that represents all messages that can be sent in two steps.
Can Bill send a message to Aaron in a maximum of two steps?
A particular website is to be designed so that all website content is available to a user within three clicks. The webpage adjacency matrix A, representing available links between different parts of the website is given.
A=\begin{bmatrix} 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 0\\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \end{bmatrix}
If every webpage can be reached within three clicks, then the matrix B = A + A^{2} + A^{3} will have no zero entries.
Find A^{2}.
Find A^{3}.
Find the matrix B= A + A^{2} + A^{3}.
Is every webpage able to be reached within three clicks?
The following map network shows the roads that connect three towns:
Create a 3 \times 3 matrix A, to represent the direct one-step paths between the three locations. For the rows and columns, use the order of Millen, Nowin then Oneslay.
Find A^{2}, which represents the two-step paths between the three locations.
How many two-step paths can be taken from Norwin to Millen?
The following map network shows connections between Joondalup, Perth, and Fremantle:
Create a 3 \times 3 matrix M, to represent the direct one-step paths between the three locations. For the rows and columns, use the order of Joondalup, Perth then Fremantle.
Find the matrix M^{2}, which represents the two-step paths between the three locations.
How many two-step paths can be taken from Freemantle to Jundalup?
