Math Problem Statement
Given A = [[8, -3], [3, y+2]], B = [[3x, z+4], [w, 1/2y]], and C = [[x, 2z], [4, y]], what is the value of matrix B if A + B = C?
Solution
To solve for the value of matrix , given that , we follow these steps:
Step 1: Understand the equation
The relationship can be rearranged as: This means we subtract from element-wise to determine .
Step 2: Write down the matrices
Given:
B = \begin{bmatrix} 3x & z+4 \\ w & \frac{1}{2}y \end{bmatrix}, \, C = \begin{bmatrix} x & 2z \\ 4 & y \end{bmatrix}$$ ### Step 3: Solve for $$B$$ Perform the subtraction $$B = C - A$$: $$\begin{bmatrix} 3x & z+4 \\ w & \frac{1}{2}y \end{bmatrix} = \begin{bmatrix} x & 2z \\ 4 & y \end{bmatrix} - \begin{bmatrix} 8 & -3 \\ 3 & y+2 \end{bmatrix}$$ Perform the subtraction element-wise: 1. First row, first column: $$3x = x - 8$$ $$3x = x - 8 \implies x = 4$$ 2. First row, second column: $$z+4 = 2z - (-3)$$ $$z+4 = 2z + 3 \implies z = 1$$ 3. Second row, first column: $$w = 4 - 3$$ $$w = 1$$ 4. Second row, second column: $$\frac{1}{2}y = y - (y+2)$$ $$\frac{1}{2}y = y - y - 2 \implies \frac{1}{2}y = -2 \implies y = -4$$ ### Step 4: Final values of $$B$$ Substituting $$x = 4$$, $$z = 1$$, $$w = 1$$, and $$y = -4$$ into $$B$$: $$B = \begin{bmatrix} 3(4) & 1+4 \\ 1 & \frac{1}{2}(-4) \end{bmatrix} = \begin{bmatrix} 12 & 5 \\ 1 & -2 \end{bmatrix}$$ ### Final Answer: $$B = \begin{bmatrix} 12 & 5 \\ 1 & -2 \end{bmatrix}$$ Let me know if you want further clarifications or have questions! --- 1. What is the process of matrix subtraction? 2. How do you verify if $$A + B = C$$ is true after solving? 3. What are the general rules for element-wise matrix operations? 4. How would this change if $$A$$, $$B$$, or $$C$$ had different dimensions? 5. What are real-life applications of matrix addition and subtraction? **Tip**: When solving for unknowns in matrices, workAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Addition
Matrix Subtraction
Algebraic Equations
Formulas
B = C - A
Theorems
Matrix Arithmetic Properties
Suitable Grade Level
Grades 9-12