Math Problem Statement
Solve sum no 7
Solve for x in the determinant equation for the 3x3 matrix provided in Problem 7.
Solution
To solve Problem 7, let's calculate the determinant of the given matrix and find the values of that satisfy the condition of the determinant being zero.
The matrix is:
x+1 & x+2 & 3 \\ 3 & x+2 & x+1 \\ x+2 & 2 & x+3 \\ \end{vmatrix} = 0$$ ### Step-by-Step Solution 1. **Calculate the Determinant:** Using cofactor expansion along the first row, we have: $$\text{det} = (x+1) \begin{vmatrix} x+2 & x+1 \\ 2 & x+3 \end{vmatrix} - (x+2) \begin{vmatrix} 3 & x+1 \\ x+2 & x+3 \end{vmatrix} + 3 \begin{vmatrix} 3 & x+2 \\ x+2 & 2 \end{vmatrix}$$ 2. **Evaluate Each Minor:** - For the first minor: $$\begin{vmatrix} x+2 & x+1 \\ 2 & x+3 \end{vmatrix} = (x+2)(x+3) - (x+1)(2) = x^2 + 5x + 6 - 2x - 2 = x^2 + 3x + 4$$ - For the second minor: $$\begin{vmatrix} 3 & x+1 \\ x+2 & x+3 \end{vmatrix} = 3(x+3) - (x+1)(x+2) = 3x + 9 - (x^2 + 3x + 2) = -x^2 + 7$$ - For the third minor: $$\begin{vmatrix} 3 & x+2 \\ x+2 & 2 \end{vmatrix} = 3(2) - (x+2)(x+2) = 6 - (x^2 + 4x + 4) = -x^2 - 4x + 2$$ 3. **Substitute and Simplify:** Substitute each minor back into the cofactor expansionAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrices
Formulas
Determinant of a 3x3 matrix using cofactor expansion
Cofactor expansion formula
Theorems
Cramer's Rule
Properties of Determinants
Suitable Grade Level
Grades 11-12