Find the value of k for which the matrix A has rank 2 7 6 1

Find the value of k for which the matrix:

A =

has rank 2

Solution

First 2 rows are clearly linearly independent

So for the matrix to have rank 2 it must lie in the span of first 2 rows

So,

(4,-3,k)=a(-7,-6,-1)+b(7,-5,12)

4=-7a+7b

-3=-6a-5b

k=-a+12b

Solving first two equations

a=1/77,b=45/77

k=-a+12b=(-1+12*45)/77=7