A) It is inconsistent
B) It has only single solution \[\operatorname{x}=0,\,\,y=0,z=0\]
C) Determinant of coefficient of matrix is zero
D) It has infinitely many solutions
Correct Answer: B
Solution :
The given system of equations obviously has \[\operatorname{x} = 0,\,\,y= 0,\,\,z =0\] as a solution. Now, to check for the presence of non-trivial solution, \[\left| \begin{matrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \\ \end{matrix} \right|\] \[=\,\,1.\left\{ 9-5\left( -1 \right) \right\}-1\left\{ 4.3-\left( -1 \right).3 \right\}+1\left\{ 4.5-3.3 \right\}\] \[=14-1(15)+\left\{ 11 \right\}=10\ne 0\] Determinant of coefficient matrix is not zero. Since solution exists (unique) for the system of equations, it is not inconsistent. So, the system of equation has only single solution, \[x=0,\,\,y=0,\,\,z=0.\]
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