https://file.aiquickdraw.com/mathbot/file/1727399639347danp9xmj.jpg

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?

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**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.

Given matrices A, B, and C such that A = [[2x, 3], [-2, 2]], B = [[4, 5], [2, x]], and C = [[2y, 26], [-2y, -6]], find the values of x and y such that A * B = C. Then, calculate the value of 2x^2 - y.

Learn how to solve a problem involving matrix multiplication with given matrices A, B, and C, and find the values of x and y. This step-by-step guide covers solving simultaneous equations using matrix operations and is suitable for students in Grades 9-12.

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