Let's break down the problem based on the image. The matrices given are:

$A = \begin{bmatrix} 1 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & -3 \\ -1 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 6 & 4 & 0 \\ 8 & 2 & -10 \end{bmatrix}$ $D = \begin{bmatrix} 3 & 5 \\ 1 & 2 \\ 0 & -4 \end{bmatrix}, \quad E = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 3 \end{bmatrix}$

The tasks are:

1. Calculate $C + D^T$ where $D^T$ is the transpose of matrix $D$.
2. Scalar multiplication $\frac{1}{3} C$.
3. Matrix multiplication $AB$.
4. Find the inverse of matrix $B$, $B^{-1}$.
5. Calculate the determinant of matrix $E$.

Let me go step by step and solve these.Here are the solutions for the given tasks:

1. $C + D^T$: $\begin{bmatrix} 9 & 5 & 0 \\ 13 & 4 & -14 \end{bmatrix}$

2. $\frac{1}{3} C$: $\begin{bmatrix} 2 & 1.33 & 0 \\ 2.67 & 0.67 & -3.33 \end{bmatrix}$

3. Matrix multiplication $AB$: $\begin{bmatrix} 0 & 7 \end{bmatrix}$

4. Inverse of matrix $B$, $B^{-1}$: $\begin{bmatrix} 0.71 & 0.43 \\ 0.14 & 0.29 \end{bmatrix}$

5. Determinant of matrix $E$: $\text{Determinant of } E = -2.0$

Let me know if you'd like further explanation or details on any part of this.

Related questions:

1. What are the properties of the transpose of a matrix?
2. How does scalar multiplication affect each element of a matrix?
3. Why does the inverse of a matrix matter, and how is it calculated?
4. What does the determinant of a matrix signify geometrically?
5. How can matrix multiplication be interpreted in terms of linear transformations?

Tip:

Always verify if a matrix is invertible by checking if its determinant is non-zero before calculating the inverse.