Math Problem Statement

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]].

Solution

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

We have three matrices AA, BB, and CC:

A=[2x322],B=[452x],C=[2y262y6]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 AB=CA \cdot B = C implies matrix multiplication between AA and BB, and their product should equal matrix CC. We are tasked with finding the value of 2x2y2x^2 - y.

Step 1: Perform matrix multiplication ABA \cdot B

To multiply matrices AA and BB, we'll follow the rule of matrix multiplication: the entry in the (i,j)(i,j)-th position of the product is obtained by taking the dot product of the ii-th row of matrix AA with the jj-th column of matrix BB.

= \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.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Systems of Equations
Algebra

Formulas

Matrix Multiplication: (A * B)_{ij} = sum(A_{ik} * B_{kj})
System of equations from matrix equality: equate elements of matrices after multiplication

Theorems

Matrix Multiplication Rules
Equality of Matrices

Suitable Grade Level

Grades 10-12