Answer to Question #298579 in Linear Algebra for shahana

Question

determine whether the homogeneous system has nontrivial solutions by inspection (without pencil and paper).

 2x1 − 3x2 + 4x3 − x4 = 0,

7x1 + x2 − 8x3 + 9x4 = 0,

2x1 + 8x2 + x3 − x4 = 0

Answer 1

Theorem

A homogeneous system always has a nontrivial solution if the number of equations is less than the number of unknowns.

We have a homogeneous system. There are three equations. There are four unknowns.

Since the number of equations is less than the number of unknowns, then by the theorem

the given homogeneous system has a nontrivial solution.