Assignment 18: Matrix equations and matrix algebra

1.

For the two matrices below, find the dimension of the solution space of $A {\mathbf x}= {\mathbf 0}$. Then find all the solutions. You should be able to determine the dimension without finding the solutions.

$A = \begin{pmatrix}1&2&0\\ 0&0&1\\ 0&0&0\end{pmatrix}$ and $A = \begin{pmatrix}1&2&0&0&3\\ 0&0&1&0&0\\ 0&0&0&1&2\end{pmatrix}$

2.

Suppose that the matrix $A$ has row-reduced echelon form $R$ given below.

$A = \begin{pmatrix}1&2&3&4&5\\ 6&7&8&9&1\\ 2&3&4&5&6\\ 7&8&9&1&2\end{pmatrix}$ and $R = \begin{pmatrix}1&0&-1&0&0\\ 0&1&2&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{pmatrix}$

Find all solutions to the matrix equation $A{\mathbf x}= \begin{pmatrix}6\\ 7\\ 8\\ 9\end{pmatrix}$ given that ${\mathbf x}= \begin{pmatrix}1\\ 0\\ 0\\ 0\\ 1\end{pmatrix}$ is a particular solution.

3.

Let $A = \begin{pmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{pmatrix}$, $B = \begin{pmatrix}1&2&3\\ 4&5&6\end{pmatrix}$, and $C = \begin{pmatrix}1&2\\ 3&4\\ 5&6\end{pmatrix}$.

Determine which of the nine products $A^2, AB, AC, BA, B^2, BC, CA, CB, C^2$ are defined. Evaluate the first three valid products.

4.

Find $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}^{1234}$. You may want to find $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}^{2}$ and $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}^{3}$ to start. Can you prove your result?

5.

Find two $2 \times 2$ matrices $A$ and $B$ neither of which is the zero matrix, but for which $AB = {\mathbf 0}$.