Consider the matrix A=\begin{bmatrix} -6 & 1 & -3 \\ 4 & -9 & 3 \end{bmatrix}. Is it possible to find A^{2}? Explain your answer.
Consider the matrix A=\begin{bmatrix} -6 & -3 \\ 1 & 4 \end{bmatrix}. Is it possible to find A^{2}? Explain your answer.
Determine whether the following matrices can be squared:
\begin{bmatrix} 6 & 18 \\ 16 & 15 \end{bmatrix}
\begin{bmatrix} 3 \\ -7 \end{bmatrix}
Determine whether the following matrices can be cubed:
\begin{bmatrix} -6 & -2 & -4 \\ -9 & 3 & 8 \\ 10 & -8 & -7 \end{bmatrix}
\begin{bmatrix} -8 \\ 4 \\ -2 \end{bmatrix}
For each of the following matrices find A^{2}:
A=\begin{bmatrix} 8 & 3 \\ 7 & 5 \end{bmatrix}
A=\begin{bmatrix} -4 & 4 \\ 9 & -3 \end{bmatrix}
A=\begin{bmatrix} 3 & 6 & 4 \\ 2 & 5 & 4 \\ 6 & 5 & 2 \end{bmatrix}
A=\begin{bmatrix} -5 & -2 & 1 \\ 3 & 4 & 4 \\ 5 & 3 & -4 \end{bmatrix}
Find the missing element in the given matrices:
{\begin{bmatrix} 5 & 2 \\ 6 & -4 \end{bmatrix}}^2 = \begin{bmatrix} 37 & 2 \\ 6 & ⬚ \end{bmatrix}
{\begin{bmatrix} 3 & 6 \\ -1 & -4 \end{bmatrix}}^2 = \begin{bmatrix} 3 & -6 \\ 1 & ⬚ \end{bmatrix}
Given A=\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, find A^{3}.
Consider the matrix {\begin{bmatrix} a & b \\ c & d \end{bmatrix}}^2.
Determine whether the following is equal to the above matrix:
\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix}
2 \begin{bmatrix} a & b \\ c & d \end{bmatrix}
\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} b & a \\ d & c \end{bmatrix}
Find the expression that would end up in the position marked X below after squaring the matrix:
{\begin{bmatrix} a & b \\ c & d \end{bmatrix}}^2 = \begin{bmatrix} X & ⬚ \\ ⬚ & ⬚ \end{bmatrix}
Consider the matrix {\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}}^2.
Determine whether the following is equal to the above matrix:
\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}
2 \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}
\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} c & b & a \\ f & e & d \\ i & h & g \end{bmatrix}
Find the expression that would end up in the position marked X below after squaring the matrix:
{\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}}^2 = \begin{bmatrix} X & ⬚ & ⬚ \\ ⬚ & ⬚ & ⬚ \\ ⬚ & ⬚ & ⬚ \end{bmatrix}
Determine the expression that goes in place of X in this equation: {\begin{bmatrix} x & y \\ z & p \end{bmatrix}}^2 = \begin{bmatrix} ⬚ & X \\ ⬚ & ⬚ \end{bmatrix}
Solve the matrix equation: {\begin{bmatrix} y & 11 \\ 4 & -2 \end{bmatrix}}^2 = \begin{bmatrix} 53 & -55 \\ -20 & 48 \end{bmatrix} for y.
Solve the matrix equation: {\begin{bmatrix} 2 & x \\ 7 & 3 \end{bmatrix}}^2 = \begin{bmatrix} 39 & 25 \\ 35 & 44 \end{bmatrix} for x.
Use your CAS calculator to find the indicated power:
A = \begin{bmatrix} 6 & 4 \\ -8 & 7 \end{bmatrix}, find A^{2}.
A = \begin{bmatrix} -5 & 3 \\ 2 & -1 \end{bmatrix}, find A^{7}.
A = \begin{bmatrix} 0.68 & 0.24 \\ 0.32 & 0.76 \end{bmatrix}, find A^{20}.
A = \begin{bmatrix} 10 & -8 & -1 \\ -5 & 1 & 2 \\ 9 & 0 & 7 \end{bmatrix}, find A^{3}.
A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, find A^{18}.
A = \begin{bmatrix} 9 & 1 & 0 & 3 \\ 0 & -2 & -1 & -4 \\ -3 & 0 & 3 & 2 \\ 5 & 0 & 2 & -7 \end{bmatrix}, find A^{4}.
A = \begin{bmatrix} 2 & 1 & 2 & 1 & 0 \\ 0 & 1 & 0 & 4 & 3 \\ 3 & 4 & 0 & 2 & 2 \\ 2 & 1 & 2 & 0 & 3 \\ 0 & 0 & 5 & 1 & 1 \end{bmatrix}, find A^{6}.
A = \begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{bmatrix}, find A^{17}.
Use your CAS calculator to find the indicated power. Round each element to 3 decimal places.
A = \begin{bmatrix} 1.01 & 0 & 0.04 \\ 0 & 0.75 & 0.21 \\ 0.44 & 0.34 & 0.24 \end{bmatrix}, find A^{25}.
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}, find A^{15}.
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} , find A^{27}.
For each matrix A, given below:
The map network below shows four towns and the paths connecting them:
The matrix A represents all of the single-step paths between the towns. Both the rows and columns are in the order: Kingston, Ashland, Greenville and then Dunham.
A = \begin{bmatrix} 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix}Use your CAS calculator to calculate A^{4}, the matrix that represents all possible four-step paths between the towns.
How many four-step paths can be taken from Ashland 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. The table below shows the possible communications:
| Andrew | Ben | Clara | Darrel | Eleanor | |
|---|---|---|---|---|---|
| Andrew | 0 | 1 | 1 | 0 | 0 |
| Ben | 0 | 0 | 1 | 1 | 0 |
| Clara | 0 | 0 | 0 | 1 | 1 |
| Darrel | 1 | 0 | 0 | 0 | 1 |
| Eleanor | 1 | 1 | 0 | 0 | 0 |
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 below.
If every webpage can be reached within three clicks, then the matrix B_3 = A + A^{2} + A^{3} will have no zero entries.