Math Problem Statement
¿Cuál es el valor de x tal que det(A) = -51?, donde A = [ [x-2, 4, 2], [-2, 1, 3], [x-4, 3, -2] ]
Solution
This problem requires finding the value of such that the determinant of the matrix is equal to . Let's calculate the determinant and solve for .
The matrix is:
x-2 & 4 & 2 \\ -2 & 1 & 3 \\ x-4 & 3 & -2 \end{bmatrix}$$ The determinant of $$A$$ can be expanded as: $$\det(A) = (x-2) \det\begin{bmatrix} 1 & 3 \\ 3 & -2 \end{bmatrix} - 4 \det\begin{bmatrix} -2 & 3 \\ x-4 & -2 \end{bmatrix} + 2 \det\begin{bmatrix} -2 & 1 \\ x-4 & 3 \end{bmatrix}$$ ### Step 1: Compute the sub-determinants 1. $$\det\begin{bmatrix} 1 & 3 \\ 3 & -2 \end{bmatrix} = (1)(-2) - (3)(3) = -2 - 9 = -11$$ 2. $$\det\begin{bmatrix} -2 & 3 \\ x-4 & -2 \end{bmatrix} = (-2)(-2) - (3)(x-4) = 4 - 3(x-4) = 4 - 3x + 12 = 16 - 3x$$ 3. $$\det\begin{bmatrix} -2 & 1 \\ x-4 & 3 \end{bmatrix} = (-2)(3) - (1)(x-4) = -6 - (x-4) = -6 - x + 4 = -2 - x$$ ### Step 2: Expand the determinant $$\det(A) = (x-2)(-11) - 4(16 - 3x) + 2(-2 - x)$$ Simplify each term: 1. $$(x-2)(-11) = -11x + 22$$ 2. $$-4(16 - 3x) = -64 + 12x$$ 3. $$2(-2 - x) = -4 - 2x$$ Combine these terms: $$\det(A) = -11x + 22 - 64 + 12x - 4 - 2x$$ $$\det(A) = (-11x + 12x - 2x) + (22 - 64 - 4)$$ $$\det(A) = -x - 46$$ ### Step 3: Solve for $$x$$ when $$\det(A) = -51$$ $$-x - 46 = -51$$ $$-x = -51 + 46$$ $$-x = -5 \implies x = 5$$ ### Final Answer: The value of $$x$$ is **5** (Option d). --- Would you like a detailed walkthrough of any step? Here are five related questions to expand your understanding: 1. How can we use cofactor expansion to calculate determinants for larger matrices? 2. Why does setting $$\det(A) = -51$$ help us solve for $$x$$? 3. What are the properties of determinants that simplify matrix calculations? 4. How does a zero determinant relate to the invertibility of a matrix? 5. What happens if multiple values of $$x$$ satisfy $$\det(A) = k$$? **Tip:** Always double-check signs when calculating determinants, as a small error can lead to incorrect results!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Formulas
Determinant calculation using cofactor expansion
Theorems
Properties of determinants
Suitable Grade Level
Grades 10-12