I have to determine all values of h for which A is invertible and I really don't know what should be my first step( If anyone could guide me through this that would be awesome. Here's the matrix:
1 1 0 1
1 0 0 1
0 1 2h + 1
0 1 1 h
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.
One more "row operation" would be to swap the third and fourth rows.
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
remains unchanged.
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.