GATE Overflow Test Series | Mixed Subjects | Test 4 | Question: 20

Question

Consider the following system of equations
$2x_{1} + 3x_{2} + 5x_{3} + 7x_{4} = 0$
$11x_{1} + 13x_{2} + 17x_{3} + 19x_{4} = 0$
$23x_{1} + 29x_{2} + 31x_{3} + 37x_{4} = 0$
$41x_{1} + 43x_{2} + 47x_{3} + 53x_{4} = 0$
The system has _________  (Mark all the appropriate choices)

• A. a trivial solution
• B. no solution
• C. a non-trivial solution
• D. a unique solution

Answer 1

We know that $AX = O$ is called system of homogeneous equation.

$\implies$ Homogeneous equations are always consistent. Because $\rho(A) = \rho(A \mid B).$

$\textbf{Case 1:}\;\text{If}\;\mid A \mid _{n \times n} \neq 0,$ then the system of equation has a unique solution, (or) trivial solution, (or) zero solution where all variables are assigned the value $0.$


$\textbf{Case 2:}\;\text{If}\;\mid A \mid _{n \times n}  =  0,$ then the system of equations have infinite number of solutions, (or) non-trivial solution.

Now, $A = \begin{bmatrix}
2 & 3 & 5 & 7 \\
11 & 13 & 17 & 19 \\
23 & 29 & 31 & 37 \\
41 & 43 & 47 & 53 \\

\end{bmatrix}$

$\implies \mid A \mid  = \begin{bmatrix}
2 & 3 & 5 & 7 \\
11 & 13 & 17 & 19 \\
23 & 29 & 31 & 37 \\
41 & 43 & 47 & 53 \\
\end{bmatrix} = 880 \neq 0.$

So, the correct answer is $A;D.$