Question

Determine if the system has a nontrivial solution. Try to use as few row operations as possible.

4x₁-6x₂ + 13x₃ = 0
-4x₁-10x₂-x₃ = 0
8x₁+4x₂ +14x₃ = 0

Answer 1

To determine if the system has a nontrivial solution, we can convert the system of equations into an augmented matrix and use row operations to solve for the variables. The system has infinitely many solutions.

Step-by-step explanation:

To determine if the system has a nontrivial solution, we can convert the system of equations into an augmented matrix and use row operations to solve for the variables.

The augmented matrix is:

• [4 -6 13 | 0]
• [-4 -10 -1 | 0]
• [8 4 14 | 0]

Using row operations, we can reduce the augmented matrix into row-echelon form or reduced row-echelon form. If the resulting matrix has a row of the form [0 0 ... 0 | c] where c is non-zero, then the system has no solution. If there are any free variables in the system, then the system has infinitely many solutions.

In this case, applying row operations, we get the matrix:

• [4 -6 13 | 0]
• [0 2 12 | 0]
• [0 0 0 | 0]

Since the last row has all zeroes except for the rightmost column, the system has a nontrivial solution. Therefore, the system has infinitely many solutions.