Describe all solutions of Ax=0

Question

Let A= [1 -4 5  4

           2 -8 10 8]

 

Describe all solutions of Ax=0

x=x₂ ?+x₃ ? +x₄?

 

Answer 1

Hi Dalia,

 

With A as the matrix given, the vector x will be the vertical vector with components {x₁, x₂, x₃, x₄}. When you multiply the two together, you'll get two equations which both must equal zero.

 

The first row times the x vector gives

1x₁ - 4x₂ + 5x₃ + 4x₄ = 0

 

And the second

2x₁ - 8x₂ + 10x₃ + 8x₄ = 0

 

You may notice that the second equation is equivalent to the first, just doubled. Therefore, it doesn't provide any additional information for your solution. With the first equation, you can solve for x₁, and you'll find that

x₁ = 4x₂ - 5x₃ - 4x₄

 

If the two equations were independent, you could have related two of the variables to the other two, for one less degree of freedom.