Grade 12 Vectors help please, intersection of planes?

Question

For what value of k will the following set of planes intersect in a line?

x-2y-z=0
x+9y-5z=0
kx-y+z=0


I have to solve this using matrices. Thank you

Answer 1

Given equations are x-2y-z=0

                                 x+9y-5z=0

                                 kx-y+z=0

The above equations can be written in matrix form as

R2 -> R2 - R1

R1 -> 11R1 + 2R2

R1 -> R1/11 and R2 -> R2/11

R3 -> R3 + R2

R3 -> R3 - KR1

In order to have a unique solution (without parameters), we need toequalise the bottom right corner to 1. We get,

                      7/11+(19/11)k=1

                      (19/11)k=1-(7/11)

                      (19/11)k=4/11

                     19k=4

                      k=4/19

Therefore the value of k is 4/11.

      

Answer 2

Rule for planes intersect in a line is Rank of the matrices is 2

x-2y-z=0
x+9y-5z=0
kx-y+z=0

Rank 2 means determent is zero.