Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 8 / c / iii

Yes. The old matrix is triangular, with determinant zero and trace $0+0+1+1=2$. The upper-left two-by-two block of the new matrix has determinant zero, so the whole determinant remains zero. Its trace isThusThis verifies both change of basis invariants from part (b).