To determine which coefficient matrix represents a system of linear equations that has a unique solution, we need to examine the determinant of each matrix. A matrix has a unique solution if its determinant is non-zero.
Given the matrices:
A. [tex]\(\left[\begin{array}{ccc} 2 & 0 & -2 \\ -7 & 1 & 5 \\ 4 & -2 & 0 \end{array}\right]\)[/tex]
B. [tex]\(\left[\begin{array}{rcr} 5 & 10 & 5 \\ 4 & 1 & 4 \\ -1 & -2 & -1 \end{array}\right]\)[/tex]
C. [tex]\(\left[\begin{array}{ccc} 4 & 2 & 6 \\ 2 & 1 & 3 \\ -2 & 3 & -4 \end{array}\right]\)[/tex]
D. [tex]\(\left[\begin{array}{ccc} 6 & 0 & -2 \\ -2 & 0 & 6 \\ 1 & -2 & 0 \end{array}\right]\)[/tex]
After calculating the determinants of each matrix, we find that:
1. The determinant of matrix A is [tex]\(0\)[/tex].
2. The determinant of matrix B is [tex]\(0\)[/tex].
3. The determinant of matrix C is [tex]\(0\)[/tex].
4. The determinant of matrix D is non-zero.
Hence, matrix D represents the system of linear equations that has a unique solution.
So, the correct answer is:
D. [tex]\(\left[\begin{array}{ccc} 6 & 0 & -2 \\ -2 & 0 & 6 \\ 1 & -2 & 0 \end{array}\right]\)[/tex]
