Math Problem Statement

By evaluating the determinants, verify the addition rule for the 2×2 case: \[ \begin{vmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} & a_{22} \end{vmatrix} = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} + \begin{vmatrix} b_{11} & b_{12} \\ a_{21} & a_{22} \end{vmatrix}. \]

Solution

The problem requires verifying the addition rule for determinants of 2×22 \times 2 matrices by explicitly evaluating the determinant expressions. Let's break this down and solve it.

Expression to Verify:

a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} & a_{22} \end{vmatrix} = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} + \begin{vmatrix} b_{11} & b_{12} \\ a_{21} & a_{22} \end{vmatrix}.$$ ### Step 1: Determinant of Left-Hand Side The determinant of the left-hand side matrix is: $$\begin{vmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} & a_{22} \end{vmatrix} = (a_{11} + b_{11})a_{22} - (a_{12} + b_{12})a_{21}.$$ Simplify: $$= a_{11}a_{22} + b_{11}a_{22} - a_{12}a_{21} - b_{12}a_{21}.$$ ### Step 2: Determinants of the Right-Hand Side #### First determinant: $$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}.$$ #### Second determinant: $$\begin{vmatrix} b_{11} & b_{12} \\ a_{21} & a_{22} \end{vmatrix} = b_{11}a_{22} - b_{12}a_{21}.$$ ### Step 3: Add the Right-Hand Side Determinants Adding the two determinants: $$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} + \begin{vmatrix} b_{11} & b_{12} \\ a_{21} & a_{22} \end{vmatrix} = (a_{11}a_{22} - a_{12}a_{21}) + (b_{11}a_{22} - b_{12}a_{21}).$$ Simplify: $$= a_{11}a_{22} + b_{11}a_{22} - a_{12}a_{21} - b_{12}a_{21}.$$ ### Step 4: Verify Equality The left-hand side and the right-hand side simplify to the same expression: $$a_{11}a_{22} + b_{11}a_{22} - a_{12}a_{21} - b_{12}a_{21}.$$ ### Conclusion The addition rule for the determinant of a $$2 \times 2$$ matrix is verified. --- Would you like further explanation on any part? Here are 5 related questions to expand on this concept: 1. How does the determinant formula extend to larger matrices? 2. What are the properties of determinants that allow this rule to hold? 3. Can this addition rule be generalized to $$n \times n$$ matrices? 4. What are the geometric interpretations of determinants in $$2 \times 2$$ matrices? 5. How does this rule relate to matrix addition and scalar multiplication? **Tip:** Remember that determinants have linearity properties, which are useful for simplifying complex matrix expressions.

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Math Problem Analysis

Mathematical Concepts

Determinants
Matrix Algebra
Linearity of Determinants

Formulas

Determinant of a 2x2 matrix: \( \text{det} \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \)
Addition property of determinants for specific configurations: \( \text{det}(A + B) = \text{det}(A) + \text{det}(B) \) (under special cases)

Theorems

Linearity Property of Determinants

Suitable Grade Level

Undergraduate Math or Advanced High School (Grades 11-12)