https://file.aiquickdraw.com/mathbot/file/1727412834488ijqxhoj1.jpg

Let's break down the given problem:

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

The problem requires us to find the values of \( x \) and \( y \) such that the matrix multiplication \( A \cdot B = C \) holds, and then determine the value of \( 2x^2 - y \).

### Step 1: Perform the matrix multiplication \( A \cdot B \)

Using the standard rule for matrix multiplication:
\[
A \cdot B = \begin{bmatrix} (2x \cdot 4 + 3 \cdot 2) & (2x \cdot 5 + 3 \cdot x) \\ (-2 \cdot 4 + 2 \cdot 2) & (-2 \cdot 5 + 2 \cdot x) \end{bmatrix}
\]
This simplifies to:
\[
A \cdot B = \begin{bmatrix} 8x + 6 & 10x + 3x \\ -8 + 4 & -10 + 2x \end{bmatrix} = \begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix}
\]

### Step 2: Set \( A \cdot B \) equal to \( C \)

Now, equate the elements of \( A \cdot B \) to the corresponding elements of \( C \):
\[
\begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix} = \begin{bmatrix} 2y & 26 \\ -2y & -6 \end{bmatrix}
\]

From this, we get the following system of equations:
1. \( 8x + 6 = 2y \)
2. \( 13x = 26 \)
3. \( -4 = -2y \)
4. \( 2x - 10 = -6 \)

### Step 3: Solve the system of equations

1. From equation 3: \( -4 = -2y \), solving for \( y \):
   \[
   y = 2
   \]
2. From equation 2: \( 13x = 26 \), solving for \( x \):
   \[
   x = 2
   \]
3. Substitute \( x = 2 \) into equation 1: 
   \[
   8(2) + 6 = 2y \implies 16 + 6 = 2y \implies 22 = 2y \implies y = 2
   \]
4. Verify with equation 4: \( 2(2) - 10 = -6 \), which is true.

### Step 4: Find the value of \( 2x^2 - y \)

Now, substitute \( x = 2 \) and \( y = 2 \) into \( 2x^2 - y \):
\[
2(2)^2 - 2 = 2(4) - 2 = 8 - 2 = 6
\]

Thus, the value of \( 2x^2 - y \) is \( \boxed{6} \).

---

Would you like more details on any of the steps? Here are 5 related questions to further explore this topic:

1. How does matrix multiplication work for non-square matrices?
2. What are the properties of a matrix inverse and when can it be applied?
3. How do you solve systems of linear equations using matrices?
4. What is the determinant of a matrix, and how is it useful in solving equations?
5. How can eigenvalues and eigenvectors be used in matrix-related problems?

**Tip:** Always double-check the dimensions of the matrices before performing multiplication to ensure that the operation is valid.

Matrix multiplication problem where A.B = C, and you need to find the values of x and y, and then calculate 2x^2 - y.

Solve the matrix multiplication problem A.B = C by finding the values of x and y, and calculate 2x^2 - y. This problem involves matrix algebra and linear equations, suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation