Given the system $\left\{\begin{array}{l}3 x+5 y+z=3 \\ 2 x+y-z=2 \\ x-3 y-4 z=1\end{array}\right.$, which statement is true?
A. The system has one unique solution because the determinant of the coefficient matrix is not zero.
B. The system has one unique solution because the determinant of the coefficient matrix is zero.
C. The system has no unique solution because the determinant of the coefficient matrix is not zero.
D. The system has no unique solution because the determinant of the coefficient matrix is zero.
Answer (1)
The system has one unique solution because the determinant of the coefficient matrix is not zero.
Explanation
Problem Analysis We are given a system of three linear equations with three unknowns, and we need to determine if the system has a unique solution based on the determinant of the coefficient matrix. The system is:
{ 3 x + 5 y + z = 3 2 x + y − z = 2 x − 3 y − 4 z = 1
The coefficient matrix A is:
A = [ 3 5 1 2 1 − 1 1 − 3 − 4 ]
We need to calculate the determinant of this matrix to determine if the system has a unique solution.
Determinant Calculation The determinant of the coefficient matrix A is calculated as follows:
det ( A ) = 3 1 − 1 − 3 − 4 − 5 2 − 1 1 − 4 + 1 2 1 1 − 3
det ( A ) = 3 ( 1 × ( − 4 ) − ( − 1 ) × ( − 3 )) − 5 ( 2 × ( − 4 ) − ( − 1 ) × 1 ) + 1 ( 2 × ( − 3 ) − 1 × 1 )
det ( A ) = 3 ( − 4 − 3 ) − 5 ( − 8 + 1 ) + 1 ( − 6 − 1 )
det ( A ) = 3 ( − 7 ) − 5 ( − 7 ) + 1 ( − 7 )
det ( A ) = − 21 + 35 − 7
det ( A ) = 7
The determinant of the coefficient matrix is 7.
Conclusion Since the determinant of the coefficient matrix is 7, which is not zero, the system of linear equations has a unique solution. A non-zero determinant indicates that the matrix is invertible, and therefore the system has a unique solution.
Final Answer The system has one unique solution because the determinant of the coefficient matrix is not zero.
Examples
Consider a scenario where you need to determine the exact amounts of three ingredients to create a specific mixture. If the relationships between the ingredients can be expressed as a system of linear equations, the determinant of the coefficient matrix helps determine if there's a unique solution. A non-zero determinant means you can find a precise combination of ingredients to achieve your desired mixture, while a zero determinant suggests that either no such combination exists or there are infinitely many possibilities.
