The problem in the image involves matrix multiplication and solving for variables. Here's a breakdown of the task:

Given:

• Matrix $A = \begin{bmatrix} 2x & 3 \\ -2 & 2 \end{bmatrix}$
• Matrix $B = \begin{bmatrix} 4 & 5 \\ 2 & x \end{bmatrix}$
• Matrix $C = \begin{bmatrix} 2y & 26 \\ -2y & -6 \end{bmatrix}$

We are asked to:

1. Compute $A \cdot B = C$, and determine the values of $x$ and $y$.
2. Calculate $2x^2 - y$.

Step 1: Matrix Multiplication $A \cdot B = C$

Multiplying matrices $A$ and $B$:

$A \cdot B = \begin{bmatrix} 2x & 3 \\ -2 & 2 \end{bmatrix} \cdot \begin{bmatrix} 4 & 5 \\ 2 & x \end{bmatrix} = \begin{bmatrix} (2x \cdot 4 + 3 \cdot 2) & (2x \cdot 5 + 3 \cdot x) \\ (-2 \cdot 4 + 2 \cdot 2) & (-2 \cdot 5 + 2 \cdot x) \end{bmatrix}$

This simplifies to: $A \cdot B = \begin{bmatrix} 8x + 6 & 10x + 3x \\ -8 + 4 & -10 + 2x \end{bmatrix} = \begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix}$

We now equate this result with matrix $C$: $\begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix} = \begin{bmatrix} 2y & 26 \\ -2y & -6 \end{bmatrix}$

Step 2: Set up equations from matrix equality

• From the first element: $8x + 6 = 2y$
• From the second element: $13x = 26$
• From the third element: $-4 = -2y$
• From the fourth element: $2x - 10 = -6$

Step 3: Solve for $x$ and $y$

From $13x = 26$: $x = \frac{26}{13} = 2$

From $-4 = -2y$: $y = \frac{-4}{-2} = 2$

Substitute $x = 2$ into $2x - 10 = -6$: $2(2) - 10 = -6 \quad \text{(True, so this is correct)}$

Now substitute $x = 2$ into $8x + 6 = 2y$: $8(2) + 6 = 2y \quad \Rightarrow \quad 16 + 6 = 2y \quad \Rightarrow \quad 22 = 2y \quad \Rightarrow \quad y = 11$

Step 4: Compute $2x^2 - y$

Substitute $x = 2$ and $y = 11$: $2x^2 - y = 2(2)^2 - 11 = 2(4) - 11 = 8 - 11 = -3$

Final Answer:

• $x = 2$
• $y = 11$
• $2x^2 - y = -3$

Would you like a detailed explanation of any part of the process?

Here are 5 questions to explore this topic further:

1. What are the steps involved in matrix multiplication?
2. How can you verify that two matrices are equal?
3. What are some common applications of matrix multiplication in real-world problems?
4. How can systems of equations be solved using matrix operations?
5. What are the properties of matrix operations (e.g., commutative, associative)?

Tip: When solving systems of equations from matrices, always check that the matrix dimensions are compatible for multiplication.