https://file.aiquickdraw.com/mathbot/file/1725535476897qv1awac3.jpg

I see the image you've uploaded with several matrix-based exercises. I'll transcribe the tasks from the image and help you solve them one by one.

Here are the exercises based on what I see:

1. If \( A = \begin{bmatrix} 6 & 2 \\ -3 & -4 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & -1 \\ -3 & 2 \end{bmatrix} \), find:
   a. \( 3A + 2B \)
   b. \( 2A - 5B \)

2. Given the matrices \( X = \begin{bmatrix} -4 & 6 \end{bmatrix} \) and \( Y = \begin{bmatrix} -6 \\ 8 \end{bmatrix} \), find \( XY \).

3. If \( M = \begin{bmatrix} 3 & -5 \\ 2 & 4 \end{bmatrix} \) and \( N = \begin{bmatrix} 2 & 5 \\ 1 & 6 \end{bmatrix} \), find \( MN \).

4. If \( D = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), find:
   a. \( D^2 \) (i.e., \( D \times D \)).

5. If \( T = \begin{bmatrix} 1 & -3 \\ -1 & 4 \end{bmatrix} \) and \( V = \begin{bmatrix} 2 & -2 \\ -1 & 3 \end{bmatrix} \), find \( TV \).

6. If \( A = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 2 \\ 5 & 1 \end{bmatrix} \), find \( AB \).

7. Given \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \\ -1 & 3 \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & -1 \end{bmatrix} \), and \( C = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \), find:
   a. \( AB - CB \)
   b. \( CA - CB \)
   c. Compare your answers from parts a and b.

I'll begin with Exercise 1. Let's calculate \( 3A + 2B \) and \( 2A - 5B \). I'll solve this using the matrix multiplication and addition formulas.For Exercise 1:

a. \( 3A + 2B = \begin{bmatrix} 22 & 4 \\ -15 & -8 \end{bmatrix} \)

b. \( 2A - 5B = \begin{bmatrix} 2 & 9 \\ 9 & -18 \end{bmatrix} \)

Would you like me to move on to the next exercises, or do you have any questions on these calculations?

Here are 5 related questions:
1. How do you compute the product of two matrices?
2. What is scalar multiplication in matrices?
3. What is the rule for matrix addition and subtraction?
4. How does matrix multiplication differ from addition?
5. Can you find the inverse of a matrix, and how?

**Tip**: In matrix operations, be cautious of matrix dimensions. For two matrices to be added or subtracted, they must have the same dimensions!

If A = [[6, 2], [-3, -4]] and B = [[2, -1], [-3, 2]], find: a. 3A + 2B b. 2A - 5B. Other exercises involve matrix multiplication and squaring, including matrix operations on various matrices.

Explore detailed solutions for matrix operations involving addition, subtraction, and multiplication. These exercises cover concepts like scalar multiplication and matrix multiplication, suitable for advanced high school or college students.

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