https://file.aiquickdraw.com/mathbot/file/1725936564062olcarp1r.jpg

Let's solve the matrix operations given in the image. The matrices \( P \), \( Q \), and \( R \) are defined as follows:

\[
P = \begin{pmatrix} 3 & -6 \\ -1 & 4 \end{pmatrix}, \quad Q = \begin{pmatrix} -1 & 9 \\ 0 & 5 \end{pmatrix}, \quad R = \begin{pmatrix} 0 & 11 \\ 13 & 12 \end{pmatrix}
\]

### a) \( 3P + Q \)
To compute \( 3P + Q \), first multiply matrix \( P \) by 3, and then add matrix \( Q \):

\[
3P = 3 \times \begin{pmatrix} 3 & -6 \\ -1 & 4 \end{pmatrix} = \begin{pmatrix} 9 & -18 \\ -3 & 12 \end{pmatrix}
\]

Now add matrix \( Q \):

\[
3P + Q = \begin{pmatrix} 9 & -18 \\ -3 & 12 \end{pmatrix} + \begin{pmatrix} -1 & 9 \\ 0 & 5 \end{pmatrix} = \begin{pmatrix} 9 + (-1) & -18 + 9 \\ -3 + 0 & 12 + 5 \end{pmatrix} = \begin{pmatrix} 8 & -9 \\ -3 & 17 \end{pmatrix}
\]

### b) \( 8P - 2Q - 3R \)
First, calculate each matrix multiplication separately:

\[
8P = 8 \times \begin{pmatrix} 3 & -6 \\ -1 & 4 \end{pmatrix} = \begin{pmatrix} 24 & -48 \\ -8 & 32 \end{pmatrix}
\]

\[
-2Q = -2 \times \begin{pmatrix} -1 & 9 \\ 0 & 5 \end{pmatrix} = \begin{pmatrix} 2 & -18 \\ 0 & -10 \end{pmatrix}
\]

\[
-3R = -3 \times \begin{pmatrix} 0 & 11 \\ 13 & 12 \end{pmatrix} = \begin{pmatrix} 0 & -33 \\ -39 & -36 \end{pmatrix}
\]

Now add all these results together:

\[
8P - 2Q - 3R = \begin{pmatrix} 24 & -48 \\ -8 & 32 \end{pmatrix} + \begin{pmatrix} 2 & -18 \\ 0 & -10 \end{pmatrix} + \begin{pmatrix} 0 & -33 \\ -39 & -36 \end{pmatrix} 
\]

\[
= \begin{pmatrix} 24 + 2 + 0 & -48 + (-18) + (-33) \\ -8 + 0 + (-39) & 32 + (-10) + (-36) \end{pmatrix} = \begin{pmatrix} 26 & -99 \\ -47 & -14 \end{pmatrix}
\]

### c) \( 2 \left( 3P - 4Q + 3R \right) \)
First, calculate the expression inside the parentheses:

\[
3P = \begin{pmatrix} 9 & -18 \\ -3 & 12 \end{pmatrix}, \quad -4Q = \begin{pmatrix} 4 & -36 \\ 0 & -20 \end{pmatrix}, \quad 3R = \begin{pmatrix} 0 & 33 \\ 39 & 36 \end{pmatrix}
\]

Now, calculate:

\[
3P - 4Q + 3R = \begin{pmatrix} 9 & -18 \\ -3 & 12 \end{pmatrix} + \begin{pmatrix} 4 & -36 \\ 0 & -20 \end{pmatrix} + \begin{pmatrix} 0 & 33 \\ 39 & 36 \end{pmatrix}
\]

\[
= \begin{pmatrix} 9 + 4 + 0 & -18 + (-36) + 33 \\ -3 + 0 + 39 & 12 + (-20) + 36 \end{pmatrix} = \begin{pmatrix} 13 & -21 \\ 36 & 28 \end{pmatrix}
\]

Finally, multiply by 2:

\[
2 \times \begin{pmatrix} 13 & -21 \\ 36 & 28 \end{pmatrix} = \begin{pmatrix} 26 & -42 \\ 72 & 56 \end{pmatrix}
\]

These are the solutions:

1. **\( 3P + Q = \begin{pmatrix} 8 & -9 \\ -3 & 17 \end{pmatrix} \)**
2. **\( 8P - 2Q - 3R = \begin{pmatrix} 26 & -99 \\ -47 & -14 \end{pmatrix} \)**
3. **\( 2(3P - 4Q + 3R) = \begin{pmatrix} 26 & -42 \\ 72 & 56 \end{pmatrix} \)**

Would you like more details or have any questions? 

Here are some related questions you might find interesting:
1. What are the properties of matrix addition and scalar multiplication?
2. How does matrix multiplication differ from addition?
3. What are the determinant and trace of a matrix?
4. How do you find the inverse of a 2x2 matrix?
5. Can you explain the concept of eigenvalues and eigenvectors?

**Tip:** Always double-check your matrix operations, especially when dealing with scalar multiplication and matrix addition/subtraction, as small sign errors can lead to incorrect results.

Solve the matrix operations involving matrices P, Q, and R: a) 3P + Q b) 8P - 2Q - 3R c) 2(3P - 4Q + 3R).

Learn how to solve matrix operations involving the matrices P, Q, and R. This problem covers matrix addition, scalar multiplication, and subtraction, along with step-by-step solutions for a) 3P + Q, b) 8P - 2Q - 3R, and c) 2(3P - 4Q + 3R). Suitable for high school students in Grades 10-12.

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