Math Problem Statement
If \( \begin{pmatrix} a & 1 - b \\ 2 - d & a \end{pmatrix} = \begin{pmatrix} 3 - b & c \\ c & 4 - d \end{pmatrix} \), then \( a + b + c + d = ? \)
Solution
To solve the given problem, we equate the matrices:
a & 1 - b \\ 2 - d & a \end{pmatrix} = \begin{pmatrix} 3 - b & c \\ c & 4 - d \end{pmatrix}.$$ From matrix equality, the corresponding elements in each position are equal. Thus, we have the following equations: 1. Top-left: $$a = 3 - b$$ 2. Top-right: $$1 - b = c$$ 3. Bottom-left: $$2 - d = c$$ 4. Bottom-right: $$a = 4 - d$$ ### Step 1: Solve for $$a$$ and $$b$$ From Equation 1: $$a = 3 - b$$. From Equation 4: $$a = 4 - d$$. Equating these two expressions for $$a$$: $$3 - b = 4 - d \implies d = b + 1.$$ ### Step 2: Solve for $$c$$ From Equation 2: $$c = 1 - b$$. From Equation 3: $$c = 2 - d$$. Equating these two expressions for $$c$$: $$1 - b = 2 - d.$$ Substitute $$d = b + 1$$ into the equation: $$1 - b = 2 - (b + 1) \implies 1 - b = 2 - b - 1 \implies 1 - b = 1 - b.$$ This confirms consistency, so $$c = 1 - b$$. ### Step 3: Calculate $$a + b + c + d$$ Now, substitute $$a = 3 - b$$, $$d = b + 1$$, and $$c = 1 - b$$: $$a + b + c + d = (3 - b) + b + (1 - b) + (b + 1).$$ Simplify: $$a + b + c + d = 3 + 1 = 4.$$ ### Final Answer: $$\boxed{4}$$ Let me know if you need further explanation or have additional questions! --- Here are some related questions for practice: 1. How do you verify if two matrices are equal? 2. What are the properties of a matrix determinant when equating two matrices? 3. How do you solve for unknowns in a system of equations derived from a matrix equality? 4. How would this problem change if the matrices had different dimensions? 5. Can you think of real-world applications of equating two matrices? **Tip:** Always equate corresponding elements in matrices to derive relationships between variables.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Equality
Equations with Variables
System of Linear Equations
Formulas
Equality of Matrices: Corresponding elements are equal
Simplification of linear equations
Theorems
Basic matrix properties
Suitable Grade Level
Grades 9-11