Math Problem Statement
Find the value of 'a' from the matrix equation:
[[a + b, 2 - a], [9, 5 - c]] = [[b + a, 2a + 8], [9, -c + 5]].
Solution
The problem involves solving the equation of two matrices being equal. When two matrices are equal, each corresponding element must also be equal.
The given matrix equation is:
= \begin{bmatrix} b + a & 2a + 8 \\ 9 & -c + 5 \end{bmatrix}.$$ ### Step 1: Compare corresponding elements 1. Top-left element: $$a + b = b + a$$ (This is trivially true since the terms are the same). 2. Top-right element: $$2 - a = 2a + 8$$. 3. Bottom-left element: $$9 = 9$$ (This is always true). 4. Bottom-right element: $$5 - c = -c + 5$$. --- ### Step 2: Solve equations #### From top-right element: $$2 - a = 2a + 8.$$ Simplify: $$2 = 2a + 8 + a,$$ $$2 = 3a + 8,$$ $$3a = -6,$$ $$a = -2.$$ #### From bottom-right element: $$5 - c = -c + 5.$$ Simplify: $$5 = -c + c + 5,$$ $$0 = 0 \quad (\text{This equation is always true and gives no additional information about } c).$$ --- ### Final Answer: The value of $$a$$ is $$\mathbf{-2}$$. Would you like a detailed explanation of any step? Here are some related questions for practice: 1. How do you solve for other variables (like $$b$$ or $$c$$) in a matrix equation? 2. What are the properties of matrix equality? 3. Can matrices of different dimensions ever be equal? 4. How can you use matrices in systems of equations? 5. What is the significance of diagonal elements in square matrices? **Tip:** Always compare each element in corresponding positions of matrices carefully when solving matrixAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equality
Linear Equations
Algebra
Formulas
Equality of corresponding elements in matrices
Theorems
Properties of Matrix Equality
Suitable Grade Level
Grades 10-12