Math Problem Statement
Evaluate the determinant of the given 3x3 matrix by forming the equivalent determinant with elements 1, 0, 0 in the first row.
Solution
To solve the determinant of the given matrix:
1 & 2 & 1 \\ 3 & 5 & -2 \\ -2 & 6 & 2 \end{vmatrix}$$ **By forming an equivalent determinant with the first row elements as $$1, 0, 0$$:** We will perform row operations to simplify the determinant while keeping the matrix equivalent. The goal is to transform the first row into $$[1, 0, 0]$$ without changing the value of the determinant. --- ### Step 1: Keep the first row unchanged $$R_1 = [1, 2, 1]$$ --- ### Step 2: Eliminate the first element of $$R_2$$ (make it 0) We use the operation: $$R_2 \to R_2 - 3R_1$$ New $$R_2$$: \[ [3, 5, -2] - 3[1, 2, 1] = [0, -1, -5] \] Updated matrix: $$\begin{bmatrix} 1 & 2 & 1 \\ 0 & -1 & -5 \\ -2 & 6 & 2 \end{bmatrix}$$ --- ### Step 3: Eliminate the first element of $$R_3$$ (make it 0) We use the operation: $$R_3 \to R_3 + 2R_1$$ New $$R_3$$: \[ [-2, 6, 2] + 2[1, 2, 1] = [0, 10, 4] \] Updated matrix: $$\begin{bmatrix} 1 & 2 & 1 \\ 0 & -1 & -5 \\ 0 & 10 & 4 \end{bmatrix}$$ --- ### Step 4: Transform the first row to $$[1, 0, 0]$$ We perform column operations to make the second and third elements of $$R_1$$ zero. #### 4.1: Eliminate the second element of $$R_1$$ We use: $$C_2 \to C_2 - 2C_1$$ New $$C_2$$: \[ [2, -1, 10] - 2[1, 0, 0] = [0, -1, 10] \] Updated matrix: $$\begin{bmatrix} 1 & 0 & 1 \\ 0 & -1 & -5 \\ 0 & 10 & 4 \end{bmatrix}$$ #### 4.2: Eliminate the third element of $$R_1$$ We use: $$C_3 \to C_3 - C_1$$ New $$C_3$$: \[ [1, -5, 4] - [1, 0, 0] = [0, -5, 4] \] Final matrix: $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & -5 \\ 0 & 10 & 4 \end{bmatrix}$$ --- ### Step 5: Evaluate the determinant The determinant of the matrix now simplifies to: $$\text{Det} = \begin{vmatrix} -1 & -5 \\ 10 & 4 \end{vmatrix}$$ Using the formula for $$2 \times 2$$ determinants: $$\text{Det} = (-1)(4) - (-5)(10) = -4 + 50 = 46$$ Thus, the determinant of the original matrix is: $$\boxed{46}$$ --- Would you like more details or clarification on any step? 😊 ### Related Questions: 1. How are determinant-preserving row and column operations defined? 2. Why do transformations simplify determinants for larger matrices? 3. What is the formula for calculating a $$3 \times 3$$ determinant directly? 4. How can row operations help in solving systems of equations? 5. What is the geometric interpretation of a determinant? ### Tip: When performing row or column operations, always track changes carefully to avoid mistakes that affect the determinant value.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Matrix Row and Column Operations
Linear Algebra
Formulas
2x2 Determinant Formula: |a b| = ad - bc
3x3 Determinant Expansion
Theorems
Properties of Determinants
Determinant-Preserving Row and Column Operations
Suitable Grade Level
Undergraduate (Linear Algebra)
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