I see that you've uploaded an image with several matrix-related questions. I'll go through each of them one by one and provide solutions.

1. If matrix A = $\begin{pmatrix} 2 & 0 \\ 3 & 4 \end{pmatrix}$ and matrix B = $\begin{pmatrix} 6 & 2 \\ -1 & 9 \end{pmatrix}$, what is the value of $A + B^T$?

We need to first find the transpose of matrix $B$ and then add it to matrix $A$.

Transpose of $B$, $B^T = \begin{pmatrix} 6 & -1 \\ 2 & 9 \end{pmatrix}$.

Now, adding matrices $A$ and $B^T$: $A + B^T = \begin{pmatrix} 2 & 0 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 6 & -1 \\ 2 & 9 \end{pmatrix} = \begin{pmatrix} 8 & -1 \\ 5 & 13 \end{pmatrix}$

So, $A + B^T = \begin{pmatrix} 8 & -1 \\ 5 & 13 \end{pmatrix}$.

2. Matrix $P = \begin{pmatrix} -2 & 0 & 2 \\ 0 & -3 & 4 \\ 0 & 0 & 6 \end{pmatrix}$. What is the determinant of matrix $P$?

Matrix $P$ is a triangular matrix (lower triangular matrix), so its determinant is the product of the diagonal elements: $\text{det}(P) = (-2) \times (-3) \times 6 = 36$

So, the determinant of $P$ is 36.

3. Give an example of a diagonal matrix.

A diagonal matrix is a square matrix in which all off-diagonal elements are zero. An example could be: [ D = \begin{pmatrix} 1 & 0 & 0 \ 0 & 2 &