Correct Answer - Option 4 : multiple solutions
Homogenous system of linear equations:
Homogenous equations are the set of equations linear equations that can be written in matrix form as AX = 0.
Where,
A is a matrix of the coefficients and X is the column matrix of the variables.
Theorem:
The system AX = 0 has,
- Unique solution (zero solution or trivial solution), if the rank of A (ρ(A) is equal to number of variables(n).
- Infinitely many solutions non-zero solutions, if ρ(A) < n.
- In this case, if |A| = 0 then the system posses infinitely many non-trivial solutions i.e., non-zero solutions.
- In this case, if |A| ≠ 0 then the system posses only a trivial solution (i.e., zero solution or unique solution.)
- If ρ(A) = r, and number of variables = n, then the number of linearly independent solution of AX = 0 is 'n-r'
Calculation:
Given that,
\( \left[\begin{matrix} 2 && -2 \\\ 1 && -1 \end{matrix}\right] \left[\begin{matrix} x_1 \\\ x_2 \end{matrix}\right]= \left[ \begin{matrix} 0 \\\ 0 \end{matrix} \right]\)
⇒ A = \( \left[\begin{matrix} 2 && -2 \\\ 1 && -1 \end{matrix}\right] \)
⇒ |A| = 0
As det A is zero, the rank of the matrix ((ρ (A) =1) < order of the matrix (n = 2).
∴ The given system posses infinitely many non-trivial solutions i.e., non-zero solutions.
