I have no clue where to go with this matrix:
1 0 k = 1
k 1 0 = 1
0 k 1 = 1
The problem says to find all values of k, if any, for which the system of linear equations has no solution.
x + kz = 1
kx + y = 1
ky + z = 1
take the determinant of to get , we want the determinant equal to zero. (Why? Can you anwer this?) so . so the matrix either has zero solutions or infinite solutions. Now how do we know which? Well note that are linearly. independent (the 3rd vector is not independent) so clearly if has a solution at all. which means , but clearly so. for k = -1, no soln. exist.