Infinite solutions of system

Question

For which values of λ does the system:

8x + 7y + 5z = 0
4x + 5y + 6z = 0
7x + 8y + λz = 0

has infinite values? Which is the way for solving these type of exercises?

Answer 1

You have a homogeneous system of equations. You can write it in matrix form as $$\begin{pmatrix}8 & 7 & 5 \\ 4 & 5 & 6\\ 7 & 8 & \lambda\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix},$$

or more compactly, as $$Ax=0,$$ where $A$ is the square matrix of the system and $x$ the vector of unknowns. Suppose that $A$ is invertible; then you can multiply by its inverse at both sides to get $$A^{-1}Ax=A^{-1}0 \Leftrightarrow x=0,$$ so in that case the system has a unique solution $(x,y,z)=(0,0,0)$.

On the other hand, if $A$ is not invertible, then $\det(A)=0$, so $\det(A-0I)=0$, which means that $A$ has a nonzero eigenvector $v$ for the eigenvalue $0$, or said otherwise, $Av=0$ with $v\neq 0$. Therefore there is at least one nonzero solution to the system. But then, if $k$ is an scalar, then $A(kv)=kAv=0$ with $k_1v\neq k_2v$ if $k_1\neq k_2$, so there is an infinite number of solutions in this case.

In conclusion, a necessary and sufficient condition for the system to have an infinite number of solutions is that $\det(A)=0$.

In your specific example, when you compute $\det(A)$ you get a polynomial in $\lambda$, which must be $0$ in order to have infinite solutions, and that gives you a way of determining the possible values of $\lambda$.

Answer 2

   A linear system may behave in any one of three possible ways:

1.The system has infinitely many solutions. 2.The system has a single unique solution. 3.The system has no solution. Any system of equations is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise there are an infinit of solutions. And for homogenous systems of equation has at least one solution, known as the zero solution (or trivial solution), which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix (det(A) ≠ 0) then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties. That is if deteterminant of matrix is zero then system has infinite solution. So only take determinant equal to zero and find value of λ.