__Homogeneous Linear Equations__:-

a_{1} x + b_{1} y = 0

a_{2} x + b_{2} y = 0

is called a system of homogeneous linear equations in x and y.

Then the above system can be written as AX = 0

__Definitions__

#### Coefficient Matrix :-

In the system AX = 0, A is called as coefficient matrix. **solution :-**

If x= x_{1}, y = y_{1} satisfy the given system AX = 0 then

**Trivial Solution :-**

For the system AX = 0, X is clearly a solution. Null matrix 0 is called zero solution or trivial solution of AX = 0.

–> If the coefficient matrix A is non-singular matrix then the system AX = 0 has only zero solution.

–> If system AX = 0 has atleast 1 non-zero solution, then coefficient matrix A is singular. This can also be written as follows:

If there exists (x, y) not all zeros, such that

a_{1} x + b_{1} y = 0

a_{2} x + b_{2} y = 0

then x, y can be eliminated from the above equations.

–> If X_{1} and X_{2} are any two solutions of the system AX = 0, then for any two scalars a_{1} and a_{2}, a_{1}X_{1} + a_{2}X_{2} is also a solution.

–> If AX = 0 has atleast 1 non-zero solution then the system has infinite number of solutions.

### Homogeneous Linear Equations in x, y, z

a_{1} x + b_{1} y + c_{1} z = 0

a_{2} x + b_{2} y + c_{2} z = 0

a_{3} x + b_{3} y + c_{3} z = 0 is called a system of homogeneous linear equations in x, y, z.

Then the above system can be written as AX = 0.

### Remarks :-

- If A is non-singular then 0 solution is the only solution of AX = 0.
- If AX = 0 has atleast one non-zero solutions then A is singular. In other words if there exists x, y, z not all zeros such that a
_{1}x + b_{1}y + c_{1}z = 0

a_{2}x + b_{2}y + c_{2}z = 0 then x, y, z can be eliminated. The eliminant is the |A| = 0.