Question

Which of the following statement is correct regarding homogeneous system?

• A. If $\left|A\right|\ne 0$ the system of equations has only trivial solution and number of solutions is one.
• B. If $\left|A\right|=0$ the system of equations has non-trival solution and the number of solutions is infinite.
• C. If the system of homogeneous linear equations has number of equations less than the number of unknowns, then it has non-trivial solution
  
• D. none of these

Answer 1

Answer: (A), (B), (C)

If the system of homogeneous linear equations has number of equations less than the number of unknowns, then it has non-trivial solution

If $\left|A\right|\ne 0$ the system of equations has only trivial solution and number of solutions is one.
If $\left|A\right|=0$ the system of equations has non-trival solution and the number of solutions is infinite.

A system of linear equations is said to be homogenous if sum of the powers of the variables in each term is same.

In other words we can say that if constant term is a zero in a system of linear equations.

Let's consider the system of linear homogeneous equations to be

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

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

$a_{3}x+b_{3}y+c_{3}z=0$

By clear observation, $x=0,y=0,z=0$ is a solution of above system of equations.

This solution is known as trivial solution.

For non-trivial solution, consider first two equations from above system.

$\frac{x}{b_1{c}_2-c_1{b}_2}=\frac{y}{c_1{a}_2-a_1{c}_2}=\frac{z}{a_1{b}_2-b_1{a}_2}=k$

$\therefore x=k(b_1{c}_2-c_1{b}_2),y=k(c_1{a}_2-a_1{c}_2),z=k(a_1{b}_2-b_1{a}_2)$

Now putting these values in the third equation, we get,

$k[a_3(b_1{c}_2-c_1{b}_2)-b_3(a_1{c}_2-c_1{a}_2)-c_3(a_1{b}_2-b_1{a}_2)]=0$

or, $a_3(b_1{c}_2-c_1{b}_2-b_3)(a_1{c}_2-c_1{a}_2)-c_3(a_1{b}_2-b_1{a}_2)=0$ where $[k\ne 0]$

This is required condition for the above system of above homogeneous linear equations to have non-trivial solution.