On pg21. of Textbook of Tensor Calculus and Differential Geometry (Nayak), I've come across the following identity: $$a^i_j A^j_k=\delta^i_k a \tag{1}$$ where $A^j_k$ are components of the cofactor matrix and a is the determinant.
I do not see how the RHS $\delta^i_k=0$ holds when $i=k$ on the LHS .
As an example let $[a^i_j]$ be a $2 \times 2$ matrix, $i,j,k=1,2$. Then by taking $i=1$ and $k=2$ we have $$a^1_1A^1_2+a^1_2A^2_2=-a^1_1a^2_1+a^1_2a^1_1 \neq \delta^1_2a=0 \tag{2}$$ Can someone please explain what has gone wrong here?