Math Problem Statement
[ − 2 − 4 8 (top row) 6 10 7 (middle row) 4 12 5 (bottom row), evaluate the determinant ∣ � ∣ ∣A∣. Use elementary row operations to evaluate |A|
Solution
To evaluate the determinant of the matrix using elementary row operations, let's start with the matrix:
-2 & -4 & 8 \\ 6 & 10 & 7 \\ 4 & 12 & 5 \end{bmatrix}$$ ### Step 1: Perform elementary row operations to simplify the matrix The goal is to simplify the matrix to a triangular form (upper or lower triangular), where the determinant is the product of the diagonal elements. Elementary row operations do not change the value of the determinant unless we swap rows or multiply by a scalar. #### Operation 1: $$R_2 \leftarrow R_2 - 3R_1$$ We subtract 3 times row 1 from row 2 to eliminate the first element of the second row: $$R_2 = 6 - 3(-2) = 6 + 6 = 12 \\ R_2 = 10 - 3(-4) = 10 + 12 = 22 \\ R_2 = 7 - 3(8) = 7 - 24 = -17$$ So, the matrix becomes: $$A = \begin{bmatrix} -2 & -4 & 8 \\ 0 & 22 & -17 \\ 4 & 12 & 5 \end{bmatrix}$$ #### Operation 2: $$R_3 \leftarrow R_3 - 2R_1$$ We subtract 2 times row 1 from row 3 to eliminate the first element of the third row: $$R_3 = 4 - 2(-2) = 4 + 4 = 8 \\ R_3 = 12 - 2(-4) = 12 + 8 = 20 \\ R_3 = 5 - 2(8) = 5 - 16 = -11$$ Now, the matrix is: $$A = \begin{bmatrix} -2 & -4 & 8 \\ 0 & 22 & -17 \\ 0 & 20 & -11 \end{bmatrix}$$ #### Operation 3: $$R_3 \leftarrow R_3 - \frac{10}{11} R_2$$ We scale row 2 and subtract it from row 3 to eliminate the second element of row 3. First, multiply $$R_2$$ by $$\frac{10}{11}$$: $$\frac{10}{11} \times (0, 22, -17) = (0, 20, -\frac{170}{11})$$ Now subtract this from row 3: $$R_3 = \left(0, 20, -11\right) - \left(0, 20, -\frac{170}{11}\right)$$ For the third element: $$-11 - \left(-\frac{170}{11}\right) = -11 + \frac{170}{11} = \frac{-121 + 170}{11} = \frac{49}{11}$$ The matrix is now: $$A = \begin{bmatrix} -2 & -4 & 8 \\ 0 & 22 & -17 \\ 0 & 0 & \frac{49}{11} \end{bmatrix}$$ ### Step 2: Compute the determinant The matrix is now in upper triangular form. The determinant is the product of the diagonal elements: $$\text{det}(A) = (-2) \times 22 \times \frac{49}{11}$$ Simplifying: $$\text{det}(A) = (-2) \times 2 \times 49 = -196$$ Thus, the determinant of the matrix $$A$$ is: $$\boxed{-196}$$ Would you like more details on any step, or do you have further questions? Here are 5 related questions: 1. What are elementary row operations and why do they preserve the determinant? 2. How do row swaps affect the determinant of a matrix? 3. What is the difference between a singular matrix and a non-singular matrix? 4. How does row echelon form help in solving systems of equations? 5. Can the determinant of a matrix help determine whether a system of equations has a unique solution? **Tip:** When working with row operations, always remember that multiplying a row by a constant multiplies the determinant by that constant.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Elementary Row Operations
Matrix Triangular Form
Formulas
Determinant of a 3x3 matrix formula
Row operation effects on determinants
Theorems
Determinant product rule
Properties of determinants with row operations
Suitable Grade Level
Grades 10-12
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