1 1 0 1
1 0 0 1
0 1 2 h + 1
0 1 1 h
Sorry if this is confusing the fourth column is 1,1,h+1, h
the first column is 1,1,0,0
third - 0,0,2,1
You mean then that the matrix is
I see two ways to do that. One is to use the fact that a matrix is invertible if and only if its determinant is non-zero. The other is to row-reduce this to triangular form and use the fact that a matrix is invertible if and only if, reduced to triangular form, it has no zeros on its main diagonal. Since a simple way of determining the determinant of a matrix is to reduce to triangular form, those are essentially the same.
That will give you
Now you also need to note that
1) If you "add a multiple of one row to another" the determinant of a matrix
2) if you "multiply one row by a number", the determinant of a matrix is multiplied by that number.
3) if you "swap two rows", the determinant of a matrix is multiplied by -1.
Since you have not "multiplied one row by a number", the determinant of your original matrix must be the determinant of this matrix: that is, .
The determinant of your original matrix is non-zero if and only if h is non-zero.