The question asks to compute $4A + 5B$, where the matrices $A$ and $B$ are given as follows:

$A = \begin{bmatrix} 9 \\ 9 \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$

To solve for $4A + 5B$, follow these steps:

Step 1: Multiply the matrices by their respective scalars:

• $4A = 4 \times \begin{bmatrix} 9 \\ 9 \end{bmatrix} = \begin{bmatrix} 4 \times 9 \\ 4 \times 9 \end{bmatrix} = \begin{bmatrix} 36 \\ 36 \end{bmatrix}$
• $5B = 5 \times \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 5 \times 2 \\ 5 \times -1 \end{bmatrix} = \begin{bmatrix} 10 \\ -5 \end{bmatrix}$

Step 2: Add the resulting matrices:

$4A + 5B = \begin{bmatrix} 36 \\ 36 \end{bmatrix} + \begin{bmatrix} 10 \\ -5 \end{bmatrix} = \begin{bmatrix} 36 + 10 \\ 36 - 5 \end{bmatrix} = \begin{bmatrix} 46 \\ 31 \end{bmatrix}$

Final Answer:

$4A + 5B = \begin{bmatrix} 46 \\ 31 \end{bmatrix}$

Let me know if you need more details or clarifications!

Here are 5 related questions to deepen your understanding:

1. What are the properties of scalar multiplication in matrix operations?
2. How do you know when matrix addition is possible?
3. Can you multiply matrices of different dimensions by the same scalar?
4. What happens when you add a matrix to its negation?
5. How do matrix operations extend to higher dimensions?

Tip: Always check if the dimensions of matrices are the same before adding them!