The problem in the image presents matrices $A$, $B$, and $C$ and asks for the value of the expression $A B + 2C$. Here's the data from the image:

• $A = \begin{pmatrix} 4 & -2 \\ -4 & 1 \end{pmatrix}$
• $B = \begin{pmatrix} -3 & -2 \\ 1 & 5 \end{pmatrix}$
• $C = \begin{pmatrix} 4 & -3 \\ -6 & 0 \end{pmatrix}$

We need to calculate the following: $A B + 2C$

Step-by-step solution:

1. Matrix Multiplication $A \times B$: $A B = \begin{pmatrix} 4 & -2 \\ -4 & 1 \end{pmatrix} \times \begin{pmatrix} -3 & -2 \\ 1 & 5 \end{pmatrix}$ We compute this by calculating each element of the resulting matrix.

2. Multiplying matrix $C$ by 2: $2C = 2 \times \begin{pmatrix} 4 & -3 \\ -6 & 0 \end{pmatrix} = \begin{pmatrix} 8 & -6 \\ -12 & 0 \end{pmatrix}$

3. Summing $A B$ and $2C$: Add the resulting matrices element-wise.

I'll now perform the calculations for $A B$ and the final result.Here are the detailed steps and results:

1. Matrix multiplication $A \times B$: $A B = \begin{pmatrix} -14 & -18 \\ 13 & 13 \end{pmatrix}$

2. Multiplying matrix $C$ by 2: $2C = \begin{pmatrix} 8 & -6 \\ -12 & 0 \end{pmatrix}$

3. Summing $A B$ and $2C$: $A B + 2C = \begin{pmatrix} -6 & -24 \\ 1 & 13 \end{pmatrix}$

Thus, the final result of $A B + 2C$ is: $\begin{pmatrix} -6 & -24 \\ 1 & 13 \end{pmatrix}$

Let me know if you'd like further clarifications or additional details!

Here are 5 related questions for further practice:

1. How would you calculate the determinant of matrix $A$?
2. What is the inverse of matrix $B$, if it exists?
3. How do you perform matrix multiplication when one matrix has different dimensions?
4. What would be the result of $A + B$ (element-wise addition)?
5. What is the trace of matrix $C$?

Tip: When performing matrix multiplication, remember that the number of columns in the first matrix must equal the number of rows in the second matrix.