The system of linear equations `ax+by=0,cx+dy=0`, has a non trivial solution if

Options

A. `ad-bclt0`
B. `ad-bcgt0`
C. `ac+bd=0`
D. `ad-bc=0`

Answer

To determine the condition under which the system of linear equations \( ax + by = 0 \) and \( cx + dy = 0 \) has a non-trivial solution, we can follow these steps:

### Step-by-step Solution:

1. **Understanding Non-trivial Solutions**: A system of linear equations has a non-trivial solution if the determinant of the coefficient matrix is zero. This means that the equations are dependent.

2. **Setting Up the Coefficient Matrix**: The given equations can be represented in matrix form as follows:
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \]

3. **Calculating the Determinant**: The determinant of the coefficient matrix is calculated as:
\[ \text{Determinant} = ad - bc \]

4. **Setting the Determinant to Zero**: For the system to have a non-trivial solution, we set the determinant equal to zero:
\[ ad - bc = 0 \]

5. **Rearranging the Equation**: This can be rearranged to express the condition:
\[ ad = bc \]

### Conclusion:
The system of linear equations \( ax + by = 0 \) and \( cx + dy = 0 \) has a non-trivial solution if:
\[ ad - bc = 0 \]