Math Problem Statement

Solve the following equation for x: \[ \begin{bmatrix} 3 & -4 & 7 & 0 & 6 & -2 \\ 2 & 0 & 1 & 8 & 0 & 0 \\ 3 & 4 & -8 & 3 & 1 & 2 \\ 27 & 6 & 5 & 0 & 0 & 3 \\ 3 & x & 0 & 2 & 1 & -1 \\ 1 & 0 & -1 & 3 & 4 & 0 \\ \end{bmatrix} \] Set the determinant equal to zero and solve for x.

Solution

The given problem requires solving for xx by finding the value that makes the determinant of the matrix equal to zero.

The matrix provided is:

3 & -4 & 7 & 0 & 6 & -2 \\ 2 & 0 & 1 & 8 & 0 & 0 \\ 3 & 4 & -8 & 3 & 1 & 2 \\ 27 & 6 & 5 & 0 & 0 & 3 \\ 3 & x & 0 & 2 & 1 & -1 \\ 1 & 0 & -1 & 3 & 4 & 0 \\ \end{bmatrix}$$ We are tasked with calculating the determinant of this matrix and setting it equal to zero, then solving for $$x$$. To do this, we will use cofactor expansion, focusing on one of the rows or columns to simplify the process. Let me calculate the determinant of the matrix and solve for $$x$$.The determinant of the matrix is given by: $$\text{determinant} = 3685x + 7370$$ To solve for $$x$$, we set this determinant equal to zero: $$3685x + 7370 = 0$$ Now, solving for $$x$$: $$x = -\frac{7370}{3685} = -2$$ Thus, the solution is $$x = -2$$. Would you like more details on the process or any further questions? Here are five related questions: 1. What is cofactor expansion, and how is it used to compute determinants? 2. How does setting a determinant equal to zero help solve matrix equations? 3. Can you solve determinants for matrices of higher dimensions using similar methods? 4. What is the significance of a zero determinant in a matrix system? 5. How do changes in individual elements affect the overall determinant of a matrix? **Tip:** Always check the matrix size before calculating the determinant, as the complexity increases significantly with larger matrices.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants

Formulas

Determinant of a square matrix
Setting determinant to zero to find variables

Theorems

Cramer's Rule
Properties of Determinants

Suitable Grade Level

Grades 11-12, College