Understanding Homogeneous Systems and Non-Trivial Solutions

A system of linear equations is called homogeneous if all the constant terms are zero. Such a system always has at least one solution, which is the trivial solution where all variables are equal to zero (e.g., x=0, y=0, z=0).

A homogeneous system of linear equations has a non-trivial solution (a solution where at least one variable is non-zero) if and only if the determinant of its coefficient matrix is equal to zero.

The given system of homogeneous equations is:

• $2x + y + 2z = 0$
• $x + y + 3z = 0$
• $4x + 3y + bz = 0$

Forming the Coefficient Matrix

We can write this system in matrix form $Ax = 0$, where A is the coefficient matrix, and x is the column vector of variables.

The coefficient matrix A is:

Calculating the Determinant for Non-Trivial Solution

For the system to have a non-trivial solution, the determinant of the coefficient matrix A must be zero ($|A| = 0$).

Let's calculate the determinant of A:

$\det(A) = \begin{vmatrix} 2 & 1 & 2 \\ 1 & 1 & 3 \\ 4 & 3 & b \end{vmatrix}$

Using the cofactor expansion along the first row:

$\det(A) = 2 \cdot \begin{vmatrix} 1 & 3 \\ 3 & b \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 3 \\ 4 & b \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & 1 \\ 4 & 3 \end{vmatrix}$

Calculate the $2 \times 2$ determinants:

• $\begin{vmatrix} 1 & 3 \\ 3 & b \end{vmatrix} = (1 \times b) - (3 \times 3) = b - 9$
• $\begin{vmatrix} 1 & 3 \\ 4 & b \end{vmatrix} = (1 \times b) - (3 \times 4) = b - 12$
• $\begin{vmatrix} 1 & 1 \\ 4 & 3 \end{vmatrix} = (1 \times 3) - (1 \times 4) = 3 - 4 = -1$

Substitute these values back into the determinant calculation:

$\det(A) = 2(b - 9) - 1(b - 12) + 2(-1)$

$\det(A) = 2b - 18 - b + 12 - 2$

Combine like terms:

$\det(A) = (2b - b) + (-18 + 12 - 2)$

$\det(A) = b - 8$

Solving for b

For a non-trivial solution, we must have $\det(A) = 0$.

$b - 8 = 0$

Solving for b:

$b = 8$

Therefore, the value of b must be 8 for the given homogeneous system of equations to have a non-trivial solution.

Revision Table: Homogeneous Systems

Additional Information: Trivial vs. Non-Trivial Solutions

For a system of linear equations $Ax = b$:

• If $b = 0$ (homogeneous system), the trivial solution is $x = 0$ (all variables are zero). This solution always exists. A non-trivial solution exists only if the determinant of the coefficient matrix $A$ is zero. If the determinant is non-zero, the trivial solution is the ONLY solution.
• If $b \ne 0$ (non-homogeneous system), the trivial solution $x=0$ is generally not a solution unless $b$ happens to be zero (which means it's actually a homogeneous system). The system might have a unique solution, infinite solutions, or no solution, depending on the determinant of A and the augmented matrix.