Let's go through the questions one by one.

1. Matrix Multiplication and Addition: Given matrices $A$ and $B$, calculate $AB + A$.

   • Matrix $A = \begin{pmatrix} 1 & 5 \\ 2 & -6 \\ 0 & 9 \end{pmatrix}$
   • Matrix $B = \begin{pmatrix} -1 & 0 \\ 3 & 5 \end{pmatrix}$

   First, compute $AB$ by multiplying $A$ and $B$, then add $A$ to the result.

2. Matrix Subtraction: Given matrices $A$ and $B$, calculate $A - B$.

   • Matrix $A = \begin{pmatrix} 1 & 5 \\ 2 & -6 \\ 0 & 9 \end{pmatrix}$
   • Matrix $B = \begin{pmatrix} -1 & 0 \\ 3 & 5 \end{pmatrix}$

   Perform element-wise subtraction between $A$ and $B$ if they are of compatible dimensions.

3. Reduced Row Echelon Form (RREF): Identify which of the given matrices (e.g., $C$ or $D$) are in reduced row echelon form, and explain why they meet or do not meet the criteria.

4. System of Linear Equations: Solve the system represented in matrix form by the equations given:

   • $2x_1 + 2x_2 + 3x_3 + 4x_4 = 9$
   • $2x_1 + 3x_2 + 3x_3 + 4x_4 = 7$
   • $2x_1 + 2x_2 + 4x_4 = -6$

   Use either Gaussian elimination or matrix methods to solve for $x_1, x_2, x_3,$ and $x_4$.

Would you like detailed solutions for each part? Let me know if you'd like explanations or if you have any questions.

Further Questions

1. How can you verify if a matrix multiplication was done correctly?
2. What criteria must a matrix meet to be considered in reduced row echelon form?
3. Can matrix subtraction always be performed between any two matrices?
4. Why might certain systems of equations have no solution or infinitely many solutions?
5. What is the geometric interpretation of the solution to a system of linear equations?

Tip

For systems of linear equations, using matrix representation can simplify solving through methods like Gaussian elimination or matrix inversion.