For waht values of x do the vectors -1,0,-1), (2,1,2), (1,1, x) form a basis for r3?

The values of x for which the given vectors are basis for R³ is:

We know that for a set of vectors are linearly independent if the matrix formed by these set of vectors is non-singular i.e. the determinant of the matrix formed by these vectors is non-zero.

We are given three vectors as:

(-1,0,-1), (2,1,2), (1,1, x)

The matrix formed by these vectors is:

Now, the determinant of this matrix is:

Hence,