From the image, we have the following matrices and operations:

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

The problem states that $A \cdot B = C$ and asks to determine the values of $x$ and $y$ such that this equation holds. Additionally, it asks to calculate the value of $2x^2 - y$.

Step 1: Matrix Multiplication $A \cdot B$

Multiplying matrix $A$ by matrix $B$:

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

The result of the multiplication is:

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

Simplifying the elements:

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

Step 2: Equating $A \cdot B$ with $C$

Now, equate the elements of $A \cdot B$ with matrix $C$:

$\begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix} = \begin{bmatrix} 2y & 26 \\ -2y & -6 \end{bmatrix}$

This gives us a system of equations:

1. $8x + 6 = 2y$
2. $13x = 26$
3. $-4 = -2y$
4. $2x - 10 = -6$

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

From equation (2): $13x = 26 \quad \Rightarrow \quad x = 2$

Substitute $x = 2$ into equations (1) and (4):

From equation (1): $8(2) + 6 = 2y \quad \Rightarrow \quad 16 + 6 = 2y \quad \Rightarrow \quad 22 = 2y \quad \Rightarrow \quad y = 11$

From equation (3): $-4 = -2y \quad \Rightarrow \quad y = 2$

This implies $y = 2$ consistently. Now substitute $x = 2$ and $y = 11$ into the expression $2x^2 - y$:

$2(2)^2 - 11 = 2(4) - 11 = 8 - 11 = -3$

Final Answer:

• $x = 2$
• $y = 11$
• The value of $2x^2 - y = -3$

Would you like more details, or do you have any further questions?

Related Questions:

1. How is matrix multiplication performed in general?
2. What are the common properties of matrix operations?
3. What is the determinant of matrix $A$ and $B$?
4. How can we use matrices to solve systems of linear equations?
5. How does substitution help in solving simultaneous equations?

Tip: When multiplying matrices, always ensure the dimensions match. The number of columns in the first matrix must equal the number of rows in the second.