Relation between $delta$-system

I’m reading Pavel Grinfeld’s book "Introduction to tensor analysis and the calculus of moving surfaces". I’ve reached the section where the author talks about $delta$-systems and the relations that bind one to the others (section 9.4, page 138). In particular there is an exercise I couldn’t get done: it asks to justify the relation:

$$ 3 delta^i_j = delta^{ij}_{rj} $$

(in the l.h.s. there is a $j$ at the place I assume belongs to an $r$, but it could be a simple index renaming). This expression is evaluated in four dimensions. Now after this equation the author explains that:

$$ delta^{ij}_{rs} = detbegin{bmatrix}
delta^i_r & delta^i_s
delta^j_r & delta^j_s
end{bmatrix} $$

Hence I assume that the contraction reduces to the above equation to:

$$ delta^{ij}_{rs} = detbegin{bmatrix}
delta^i_r & delta^i_j
delta^j_r & delta^j_j
end{bmatrix} = delta^i_r – delta^i_j delta^j_r = delta^i_r – delta^i_r = 0$$

But this contradicts the fact that $ 3 delta^i_j = delta^{ij}_{rj} $. What am I doing wrong? How could I solve this exercise?

P.S. Thank you for your answers in advance and excuse my poor English, I’m still practising it.