Answer
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Hint – To find the number of non-trivial solutions of given equations we write the set of equations in matrix form. Then find its determinant and equate it to 0.
Complete step-by-step answer:
For a non-trivial solution the determinant of the respective matrix = 0
$ \Rightarrow \left( {\begin{array}{*{20}{c}}
{\text{t}}&{{\text{t + 1}}}&{{\text{t - 1}}} \\
{{\text{t + 1}}}&{\text{t}}&{{\text{t + 2}}} \\
{{\text{t - 1}}}&{{\text{t + 2}}}&{\text{t}}
\end{array}} \right)$ = 0
Now, we reduce the matrix using row operations
R2 -> R2 – R1
R3 -> R3 – R1
Which gives us,
\[\left( {\begin{array}{*{20}{c}}
{\text{t}}&{{\text{t + 1}}}&{{\text{t - 1}}} \\
{\text{1}}&{ - 1}&3 \\
{ - 1}&1&1
\end{array}} \right) = 0\]
For $
{\text{A = }}\left( {\begin{array}{*{20}{c}}
{\text{a}}&{\text{b}}&{\text{c}} \\
{\text{d}}&{\text{e}}&{\text{f}} \\
{\text{g}}&{\text{h}}&{\text{i}}
\end{array}} \right) \\
\\
$, Det A = a (ei - fh) - b (di - fg) + c (dh - eg)
⟹t (-1 x 1 – 3 x 1) – (t + 1) (1 x 1 – (3 x -1)) + (t -1) (1 x 1 – (-1 x -1)) = 0
⟹t (-1 -3) – (t + 1)(1 +3) + (t – 1)(1 – 1) = 0
⟹-4t -4t -4 = 0
⟹-8t – 4 = 0
⟹t =$ - \dfrac{1}{2}$.
‘t’ has only one value for which the system has non- homogeneous equations and has non- trivial solutions. Hence Option C is the correct answer.
Note: The key point to solve such problems is to know that for a non-trivial solution the determinant of the matrix is zero.
A (n x n) homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.
Complete step-by-step answer:
For a non-trivial solution the determinant of the respective matrix = 0
$ \Rightarrow \left( {\begin{array}{*{20}{c}}
{\text{t}}&{{\text{t + 1}}}&{{\text{t - 1}}} \\
{{\text{t + 1}}}&{\text{t}}&{{\text{t + 2}}} \\
{{\text{t - 1}}}&{{\text{t + 2}}}&{\text{t}}
\end{array}} \right)$ = 0
Now, we reduce the matrix using row operations
R2 -> R2 – R1
R3 -> R3 – R1
Which gives us,
\[\left( {\begin{array}{*{20}{c}}
{\text{t}}&{{\text{t + 1}}}&{{\text{t - 1}}} \\
{\text{1}}&{ - 1}&3 \\
{ - 1}&1&1
\end{array}} \right) = 0\]
For $
{\text{A = }}\left( {\begin{array}{*{20}{c}}
{\text{a}}&{\text{b}}&{\text{c}} \\
{\text{d}}&{\text{e}}&{\text{f}} \\
{\text{g}}&{\text{h}}&{\text{i}}
\end{array}} \right) \\
\\
$, Det A = a (ei - fh) - b (di - fg) + c (dh - eg)
⟹t (-1 x 1 – 3 x 1) – (t + 1) (1 x 1 – (3 x -1)) + (t -1) (1 x 1 – (-1 x -1)) = 0
⟹t (-1 -3) – (t + 1)(1 +3) + (t – 1)(1 – 1) = 0
⟹-4t -4t -4 = 0
⟹-8t – 4 = 0
⟹t =$ - \dfrac{1}{2}$.
‘t’ has only one value for which the system has non- homogeneous equations and has non- trivial solutions. Hence Option C is the correct answer.
Note: The key point to solve such problems is to know that for a non-trivial solution the determinant of the matrix is zero.
A (n x n) homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.
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