Assignment 18: Matrix equations and matrix algebra

1. For the two matrices below, find the dimension of the solution space of Ax = 0. Then find all the solutions. You should be able to determine the dimension without finding the solutions.

A = (\begin{matrix}1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) and A = (\begin{matrix}1 & 2 & 0 & 0 & 3 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 \end{matrix})

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

A = (\begin{matrix}1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 7 & 8 & 9 & 1 & 2 \end{matrix}) and R = (\begin{matrix}1 & 0 & −1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

Find all solutions to the matrix equation Ax = (\begin{matrix}6 \\ 7 \\ 8 \\ 9 \end{matrix}) given that x = (\begin{matrix}1 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix}) is a

particular solution.

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

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{matrix}1 & 1 \\ 0 & 1 \end{matrix})^{1234}. You may want to find (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})^{2} and (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})^{3} to start. Can you prove

your result?

5. Find two 2 × 2 matrices A and B neither of which is the zero matrix, but for which AB = 0.