Math Problem Statement
If a 2x2 matrix \(\begin{pmatrix} a & 1-b \\ 2-d & a \end{pmatrix}\) is equal to \(\begin{pmatrix} 3-b & c \\ c & 4-d \end{pmatrix}\), find \(a + b + c + d\).
Solution
The problem states:
If
a & 1-b \\ 2-d & a \end{pmatrix} = \begin{pmatrix} 3-b & c \\ c & 4-d \end{pmatrix},$$ then $$a + b + c + d = ?$$. ### Solution: 1. **Equating Matrices**: Since the two matrices are equal, corresponding elements must be equal: $$a = 3-b, \quad 1-b = c, \quad 2-d = c, \quad a = 4-d.$$ 2. **Simplify Equations**: From $$a = 3-b$$, we solve for $$b$$: $$b = 3-a.$$ From $$1-b = c$$: $$c = 1-b.$$ From $$2-d = c$$: $$d = 2-c.$$ From $$a = 4-d$$: $$d = 4-a.$$ 3. **Substitute**: Substitute $$d = 4-a$$ into $$d = 2-c$$: $$4-a = 2-c \implies c = a-2.$$ Substitute $$b = 3-a$$ into $$c = 1-b$$: $$c = 1-(3-a) \implies c = a-2.$$ This confirms $$c = a-2$$ is consistent. 4. **Substitute $$c$$ into $$d$$**: From $$d = 2-c$$: $$d = 2-(a-2) \implies d = 4-a.$$ 5. **Sum $$a + b + c + d$$**: Substitute all values: $$b = 3-a, \quad c = a-2, \quad d = 4-a.$$ Sum them: $$a + b + c + d = a + (3-a) + (a-2) + (4-a).$$ Simplify: $$a + b + c + d = 3 + 4 - 2 = 5.$$ ### Final Answer: $$\boxed{5}$$ Let me know if you want further clarification! --- ### Expanding Understanding with Questions: 1. What are some real-world applications of matrix equality? 2. How would this problem change if the matrices were $$3 \times 3$$? 3. Can you explain why equating corresponding elements is valid in matrix equality? 4. How can determinants verify if two matrices are equal? 5. What role does solving simultaneous equations play in such problems? ### Tip: Always cross-check substitutions when solving simultaneous equations to avoid inconsistencies!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equality
Simultaneous Equations
Formulas
Matrix Equality (Corresponding Elements)
Substitution in Equations
Theorems
Properties of Equal Matrices
Suitable Grade Level
Grades 9-12