**SOLUTION OF HOMOGENEOUS SYSTEM OF LINEAR EQUATIONS: **Let AX = 0 be a homogeneous system of simultaneous linear equations with n-unknowns. Let us now discuss two cases:

**CASE I:** When |A| ≠ 0.

In this case, A^{-1} exists.

*AX = 0 *

*A ^{-1} (AX) = A^{-1}0 *

*(A ^{-1}A) X = 0 *

*I _{n}X = 0 *

*X = 0 *

*X _{1 }= X_{2} = … = X_{n }= 0 *

Thus, if A is a non-singular matrix the homogeneous system of equations has a unique solution given by => x_{1} = x_{2} = … = x_{n} = 0. This solution is known as a trivial solution.

**CASE II** When |A| = 0

In this case, we have (adj A) B = (adj A) 0 = 0

So, the homogenous system of equation is consistent and it has infinitely many solutions can be obtained by giving any real value to one of the variables.

It follows from the above discussion that a homogeneous system of equations given by AX=O is always consistent. It has a unique solution in which all variables are equal to zero, if |A| ≠ 0. If |A| = 0, then the system has non-trivial solutions.

**Example 1: **Solve the following system of homogeneous equations:

2x + 3y – z = 0

x – y – 2z = 0

3x + y + 3z = 0

**SOLUTION: **The given system of homogeneous equations can be written as

\(\left[ \begin{matrix}2 & 3 & -1 \\1 & -1 & -2 \\3 & 1 & 3\\\end{matrix} \right]\left[ \begin{matrix}x \\y \\z \\\end{matrix} \right]=\left[ \begin{matrix}0 \\0 \\0 \\\end{matrix} \right]\)

Or AX = 0, where \(A=\left[ \begin{matrix}2 & 3 & -1 \\1 & -1 & -2 \\3 & 1 & 3 \\\end{matrix} \right] \), \(X=\left[ \begin{matrix}x \\y \\z \\\end{matrix} \right] \) and \(O=\left[ \begin{matrix}0 \\0 \\0 \\\end{matrix} \right] \).

Now, \(\left| A \right|=\left[ \begin{matrix}2 & 3 & -1 \\1 & -1 & -2 \\3 & 1 & 3 \\\end{matrix} \right] \) = – 2 – 27 – 4 = – 33 ≠ 0.

Thus |A| ≠ 0. So, the given system has only the trivial solution given by x = y = z = O.