Answer
Verified
396.3k+ views
Hint:If we are given the system of equations like here we are given:
$x + \lambda y - z = 0$
$\lambda x - y - z = 0$
$x + y - \lambda z = 0$
If all variable values are not equal to $0$, then it is a non-trivial solution.
$\left| {\begin{array}{*{20}{c}}
1&\lambda &{ - 1} \\
\lambda &{ - 1}&{ - 1} \\
1&1&{ - \lambda }
\end{array}} \right| = 0$
Get the determinant of coefficients and find the value of $\lambda $.
Complete step-by-step answer:
So here we are given the system of equations like here we are given:
$x + \lambda y - z = 0$
$\lambda x - y - z = 0$
$x + y - \lambda z = 0$
And it is a non-trivial solution.
It means that at least one of the variables must not be equal to $0$.
If all are equal to zero, then it is a trivial solution.
So here the determinant of the coefficients as shown must be equal to $0$
$\left| {\begin{array}{*{20}{c}}
1&\lambda &{ - 1} \\
\lambda &{ - 1}&{ - 1} \\
1&1&{ - \lambda }
\end{array}} \right| = 0$
Now solving it, we get
$1(\lambda + 1) - \lambda ( - {\lambda ^2} + 1) - 1(\lambda + 1) = 0$
$(\lambda + 1) - \lambda (1 - {\lambda ^2}) - 1(\lambda + 1) = 0$
We know identity $(a-b) (a+b) = a^2-b^2$
Applying the identity to 2nd term
$(\lambda + 1) + \lambda ({\lambda ^2}-1) - 1(\lambda + 1) = 0$
$(\lambda + 1) + \lambda (\lambda + 1) (\lambda - 1) - 1(\lambda + 1) = 0$
$(\lambda + 1) + \lambda (\lambda + 1) (\lambda - 1) - 1(\lambda + 1) = 0$
Taking $(\lambda + 1)$ and simplifying further we get,
$\lambda (\lambda - 1)(\lambda + 1) = 0$
So we get
$\lambda $$ = 0,1, - 1$
So there are three values for which it is a non-trivial solution.
So, the correct answer is “Option D”.
Note:If set of the linear equations are given to be trivial then variables all are equal to zero but if it is a non-trivial solution then at least one must not be equal to zero.
$x + \lambda y - z = 0$
$\lambda x - y - z = 0$
$x + y - \lambda z = 0$
If all variable values are not equal to $0$, then it is a non-trivial solution.
$\left| {\begin{array}{*{20}{c}}
1&\lambda &{ - 1} \\
\lambda &{ - 1}&{ - 1} \\
1&1&{ - \lambda }
\end{array}} \right| = 0$
Get the determinant of coefficients and find the value of $\lambda $.
Complete step-by-step answer:
So here we are given the system of equations like here we are given:
$x + \lambda y - z = 0$
$\lambda x - y - z = 0$
$x + y - \lambda z = 0$
And it is a non-trivial solution.
It means that at least one of the variables must not be equal to $0$.
If all are equal to zero, then it is a trivial solution.
So here the determinant of the coefficients as shown must be equal to $0$
$\left| {\begin{array}{*{20}{c}}
1&\lambda &{ - 1} \\
\lambda &{ - 1}&{ - 1} \\
1&1&{ - \lambda }
\end{array}} \right| = 0$
Now solving it, we get
$1(\lambda + 1) - \lambda ( - {\lambda ^2} + 1) - 1(\lambda + 1) = 0$
$(\lambda + 1) - \lambda (1 - {\lambda ^2}) - 1(\lambda + 1) = 0$
We know identity $(a-b) (a+b) = a^2-b^2$
Applying the identity to 2nd term
$(\lambda + 1) + \lambda ({\lambda ^2}-1) - 1(\lambda + 1) = 0$
$(\lambda + 1) + \lambda (\lambda + 1) (\lambda - 1) - 1(\lambda + 1) = 0$
$(\lambda + 1) + \lambda (\lambda + 1) (\lambda - 1) - 1(\lambda + 1) = 0$
Taking $(\lambda + 1)$ and simplifying further we get,
$\lambda (\lambda - 1)(\lambda + 1) = 0$
So we get
$\lambda $$ = 0,1, - 1$
So there are three values for which it is a non-trivial solution.
So, the correct answer is “Option D”.
Note:If set of the linear equations are given to be trivial then variables all are equal to zero but if it is a non-trivial solution then at least one must not be equal to zero.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE