Activity 6.2.4

In Sage, the transpose of a matrix A is given by A.T. Define the matrices $$A = \begin{bmatrix} 1 & 0 & -3 \\ 2 & -2 & 1 \\ \end{bmatrix}, \hspace{6pt} B = \begin{bmatrix} 3 & -4 & 1 \\ 0 & 1 & 2 \\ \end{bmatrix}, \hspace{6pt} C= \begin{bmatrix} 1 & 0 & -3 \\ 2 & -2 & 1 \\ 3 & 2 & 0 \\ \end{bmatrix}\text{.}$$

1. Evaluate $(A+B)^T$ and $A^T+B^T$. What do you notice about the relationship between these two matrices?

2. What happens if you transpose a matrix twice; that is, what is $(A^T)^T$?

3. Find $\det(C)$ and $\det(C^T)$. What do you notice about the relationship between these determinants?

4. 1. Find the product $AC$ and its transpose $(AC)^T$.

   2. Is it possible to compute the product $A^TC^T$? Explain why or why not.

   3. Find the product $C^TA^T$ and compare it to $(AC)^T$. What do you notice about the relationship between these two matrices?

5. What is the transpose of the identity matrix $I$?

6. If a square matrix $D$ is invertible, explain why you can guarantee that $D^T$ is invertible and why $(D^T)^{-1} = (D^{-1})^T$.