https://file.aiquickdraw.com/mathbot/file/172778553899570sx4vkd.jpg

Let's analyze the given matrix equation from the image:

We have three matrices \( A \), \( B \), and \( C \):

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

The equation \( A \cdot B = C \) implies matrix multiplication between \( A \) and \( B \), and their product should equal matrix \( C \). We are tasked with finding the value of \( 2x^2 - y \).

### Step 1: Perform matrix multiplication \( A \cdot B \)
To multiply matrices \( A \) and \( B \), we'll follow the rule of matrix multiplication: the entry in the \( (i,j) \)-th position of the product is obtained by taking the dot product of the \( i \)-th row of matrix \( A \) with the \( j \)-th column of matrix \( B \).

\[
A \cdot B = \begin{bmatrix} (2x)(4) + (3)(2) & (2x)(5) + (3)(x) \\ (-2)(4) + (2)(2) & (-2)(5) + (2)(x) \end{bmatrix}
= \begin{bmatrix} 8x + 6 & 10x + 3x \\ -8 + 4 & -10 + 2x \end{bmatrix}
= \begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix}
\]

### Step 2: Set the product equal to matrix \( C \)
Now, equate this product to matrix \( C \):

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

From this equality, we can set up the following system of equations by comparing corresponding elements:

1. \( 8x + 6 = 2y \)
2. \( 13x = 26 \)
3. \( -4 = -2y \)
4. \( 2x - 10 = -6 \)

### Step 3: Solve the system of equations

From equation (2): 
\[
13x = 26 \implies x = 2
\]

From equation (3): 
\[
-4 = -2y \implies y = 2
\]

Now, substitute \( x = 2 \) into equation (1):

\[
8(2) + 6 = 2y \implies 16 + 6 = 2y \implies 22 = 2y \implies y = 11
\]

Thus, \( y = 11 \) and \( x = 2 \).

### Step 4: Calculate \( 2x^2 - y \)
Now we calculate \( 2x^2 - y \):

\[
2(2)^2 - 11 = 2(4) - 11 = 8 - 11 = -3
\]

So, the value of \( 2x^2 - y \) is \( -3 \).

---

Do you want more details or have any further questions?  
Here are five related questions to expand the topic:

1. How do you multiply two matrices in general?
2. What are the conditions for matrix multiplication to be valid?
3. What is the determinant of matrix \( A \)?
4. How can you verify if two matrices are equal?
5. How do matrix equations relate to systems of linear equations?

**Tip:** When solving matrix equations, always ensure the dimensions of matrices allow for multiplication.

Solve the matrix equation A * B = C and find the value of 2x^2 - y, given the matrices: A = [[2x, 3], [-2, 2]], B = [[4, 5], [2, x]], and C = [[2y, 26], [-2y, -6]].

Learn how to solve the matrix equation A * B = C and find the value of 2x^2 - y with a step-by-step solution. This problem covers key concepts like matrix multiplication, systems of equations, and algebra. Suitable for high school students (Grades 10-12).

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