Math Problem Statement

Solve the determinant equation involving a 3x3 matrix: | -1 2x+1 3 | | 4 3x 5 | | -2 -1 4 | = 3 - 5x.

Solution

This is a determinant equation involving a 3x3 matrix on the left-hand side, set equal to an expression on the right-hand side. Let's solve it step by step:

The determinant of the matrix is given by:

- (2x + 1) \cdot \begin{vmatrix} 4 & 5 \\ -2 & 4 \end{vmatrix} + 3 \cdot \begin{vmatrix} 4 & 3x \\ -2 & -1 \end{vmatrix}$$ ### Step 1: Expand each 2x2 determinant The determinant of a 2x2 matrix $$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$$ is calculated as $$ad - bc$$. 1. For $$\begin{vmatrix} 3x & 5 \\ -1 & 4 \end{vmatrix}$$: $$= (3x)(4) - (-1)(5) = 12x + 5$$ 2. For $$\begin{vmatrix} 4 & 5 \\ -2 & 4 \end{vmatrix}$$: $$= (4)(4) - (5)(-2) = 16 + 10 = 26$$ 3. For $$\begin{vmatrix} 4 & 3x \\ -2 & -1 \end{vmatrix}$$: $$= (4)(-1) - (3x)(-2) = -4 + 6x = 6x - 4$$ ### Step 2: Substitute into the determinant formula $$\text{Determinant} = -1(12x + 5) - (2x + 1)(26) + 3(6x - 4)$$ Simplify each term: 1. $$-1(12x + 5) = -12x - 5$$ 2. $$-(2x + 1)(26) = -(52x + 26) = -52x - 26$$ 3. $$3(6x - 4) = 18x - 12$$ Add these together: $$\text{Determinant} = -12x - 5 - 52x - 26 + 18x - 12$$ $$= (-12x - 52x + 18x) + (-5 - 26 - 12)$$ $$= -46x - 43$$ ### Step 3: Set equal to the right-hand side The equation is: $$-46x - 43 = 3 - 5x$$ Solve for $$x$$: 1. Combine like terms: $$-46x + 5x = 3 + 43$$ $$-41x = 46$$ 2. Solve for $$x$$: $$x = -\frac{46}{41}$$ ### Final Answer: $$x = -\frac{46}{41}$$ Would you like a detailed verification of this solution or further explanations? --- ### 5 Related Questions: 1. How do you compute determinants for higher-dimensional matrices? 2. What are some real-world applications of determinants in engineering or physics? 3. Can this equation be solved using numerical approximation instead of algebraic methods? 4. How does the determinant relate to the invertibility of a matrix? 5. What happens if $$x$$ leads to a determinant of zero? ### Tip: Always double-check your determinant expansion for sign errors, as it's easy to make mistakes with alternating signs.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants
Equation Solving

Formulas

Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Determinant of a 2x2 matrix: det = ad − bc

Theorems

Properties of Determinants

Suitable Grade Level

Undergraduate Mathematics or Advanced High School Level