Correct Answer - Option 2 : -2
Given:
2x - y - z = 2,
x - 2y + z = -4,
x + y + λz = 4
Concept:
Consider the system of m linear equations
a11 x1 + a12 x2 + … + a1n xn = b1
a21 x1 + a22 x2 + … + a2n xn = b2
…
am1 x1 + am2 x2 + … + amn xn = bm
The above equations containing the n unknowns x1, x2, …, xn. To determine whether the above system of equations is consistent or not, we need to find the rank of the following matrices.
\(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}\\ \ldots & \ldots & \ldots & \ldots \\ {{a_{m1}}}&{{a_{m2}}}& \ldots &{{a_{mn}}} \end{array}} \right]\) and \(\left[ {A{\rm{|}}B} \right] = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}&{{b_1}}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}&{{b_2}}\\ \ldots & \ldots & \ldots & \ldots & \ldots \\ {{a_{m1}}}&{{a_{m2}}}& \ldots &{{a_{mn}}}&{{b_m}} \end{array}} \right]\)
A is the coefficient matrix and [A|B] is called an augmented matrix of the given system of equations.
If the rank of matrix A is not equal to rank of augmented matrix, then system is inconsistent, and it has no solution.
Rank of A ≠ Rank of augmented matrix. Therefore if det (A) = 0 system is inconsistent (no solution)
Calculation:
\( A = \left[ {\begin{array}{*{20}{c}} 2&{ - 1}&{ - 1}\\ 1&{ - 2}&1\\ 1&1&λ \end{array}} \right]\;\;B = \left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right]\;C = \left[ {\begin{array}{*{20}{c}} 2\\ { - 4}\\ 4 \end{array}} \right] \)
Hence, for no solution, |A| = 0
\( ⇒ \left| {\begin{array}{*{20}{c}} 2&{ - 1}&{ - 1}\\ 1&{ - 2}&1\\ 1&1&λ \end{array}} \right|\; = 0 \)
⇒ 2(-2λ - 1) + 1(λ - 1) - 1(1 + 2) = 0
⇒ -4λ - 2 + λ - 1 - 3 = 0
⇒ -3λ = 6
⇒ λ = - 2
- If the rank of matrix A is equal to the rank of an augmented matrix and it is equal to a number of unknowns, then system is consistent and there is a unique solution.
- Rank of A = Rank of augmented matrix = n
- If the rank of matrix A is equal to rank of augmented matrix and it is less than the number of unknowns, then system is consistent and there are infinite number of solutions.
- Rank of A = Rank of augmented matrix < n
