Correct Answer - Option 2 : infinitely many solutions
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 following matrices.
\(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}\\ \ldots & \ldots & \ldots & \ldots \\ {{a_{m1}}}&{{a_{m2}}}& \ldots &{{a_{mn}}} \end{array}} \right]\)
\(\left[ {A{\rm{|}}B} \right] = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}&{{b_1}}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}&{{b_2}}\\ \ldots & \ldots & \ldots & \ldots & \ldots \\ {{a_{m1}}}&{{a_{m2}}}& \ldots &{{a_{mn}}}&{{b_m}} \end{array}} \right]\)
A is the coefficient matrix and [A|B] is called an augmented matrix of the given system of equations.
We can find the consistency of the given system of equations as follows:
(i) If the rank of matrix A is equal to the rank of an augmented matrix and it is equal to the number of unknowns, then the system is consistent and there is a unique solution.
The rank of A = Rank of augmented matrix = n
(ii) If the rank of matrix A is equal to the rank of an augmented matrix and it is less than the number of unknowns, then the system is consistent and there are an infinite number of solutions.
The rank of A = Rank of augmented matrix < n
(iii) If the rank of matrix A is not equal to the rank of the augmented matrix, then the system is inconsistent, and it has no solution.
The rank of A ≠ Rank of an augmented matrix
Calculation:
The given system of equations can be represented in a matrix form as shown below.
\(A = \left[ {\begin{array}{*{20}{c}} 2&1&{ 3}\\ 3&0&1\\ 1&{ 2}&{ 5} \end{array}} \right],X = \left[ {\begin{array}{*{20}{c}} a\\ b\\ c\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}} { 5}\\ { -4}\\ 14 \end{array}} \right]\)
The Augmented matrix can be written by:
\([A|B] = \left[ {\begin{array}{*{20}{c}} 2&1&{ 3}\\ 3&0&1\\ 1&{ 2}&{ 5} \end{array}{\rm{|}}\begin{array}{*{20}{c}} { 5}\\ { -4}\\ 14 \end{array}} \right]\)
R2 → 2R2 – 3R1 and R3 → 2R3 – R1
\(= \left[ {\begin{array}{*{20}{c}} 2&1&{ 9}\\ 0&-3&-7\\ 0&{ 3}&{ 7} \end{array}{\rm{|}}\begin{array}{*{20}{c}} { 5}\\ { - 23}\\ 23 \end{array}} \right]\;\)
R3 → R3 + R2
\([A|B] = \left[ {\begin{array}{*{20}{c}} 2&1&{ 9}\\ 0&-3&-7\\ 0&{ 0}&{ 0} \end{array}{\rm{|}}\begin{array}{*{20}{c}} { 5}\\ { - 23}\\ 0 \end{array}} \right]\;\)
The rank of matrix A = 2
The rank of Augmented matrix = 2
Rank of A = Rank of augmented matrix = 2 < n = 3
Hence, the system is consistent and has infinitely many solutions.
