Add a new row to datagrid view and reflect the changes in database

Question

I have a dgv which I bind to a datatable.

public Form1()
{
    da = new SqlDataAdapter("Select * from workers order by id", connStr);
    cb = new SqlCommandBuilder(da);
    dt = new DataTable();
}

Here I fill the datatable and link the gridview to the datasource

private void button1_Click(object sender, EventArgs e)
{
    dt.Clear();
    da.Fill(dt);
    dataGridView1.DataSource = dt;
}

When I want to add a new row to the datagridview I do the following:

private void button2_Click(object sender, EventArgs e)
{
    dt.Rows.Add(dt.NewRow());

    da.Update(dt);  //reflects changes to database

    dt.Clear();
    da.Fill(dt);
    dataGridView1.FirstDisplayedScrollingRowIndex = dataGridView1.Rows.Count - 1;
}

Is there a way to add the row without having to reload the whole table?