Question

The following system of linear equations
2x + 3y + 2z = 9
3x + 2y + 2z = 9
x - y + 4z = 8

• A. does not have any solution
• B. has a unique solution
• C. has infinitely many solutions
• D. has a solution ($\alpha, \beta, \gamma$) satisfying $\alpha + \beta^2 + \gamma^2 = 12$

Hint:

For a system of linear equations AX = B, the nature of the solution is determined by the determinant of the coefficient matrix A. If det(A) $\neq$ 0, the system has a unique solution. If det(A) = 0, the system may have infinitely many solutions or no solution, which requires further investigation of (adj A)B.

Answer 1

The Correct Option is B

To determine the nature of the solution for a system of linear equations, we first calculate the determinant of the coefficient matrix, denoted by $\Delta$. 

$\Delta = \det(A) = 2(2 \cdot 4 - 2 \cdot (-1)) - 3(3 \cdot 4 - 2 \cdot 1) + 2(3 \cdot (-1) - 2 \cdot 1)$. 
$\Delta = 2(8 + 2) - 3(12 - 2) + 2(-3 - 2)$. 
$\Delta = 2(10) - 3(10) + 2(-5)$. 
$\Delta = 20 - 30 - 10 = -20$. 
Since the determinant $\Delta$ is non-zero ($\Delta = -20 \neq 0$), the system of linear equations is consistent and has a unique solution. 
Therefore, option (B) is the correct statement.