Issue with Matrix Multiplication using the Formal Definition

If I understood correctly, the matrix $E = \begin{bmatrix}E_{11} & E_{12} \\ E_{21} & E_{22}\end{bmatrix}$ is given by

$$E = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$$

so for $1\le i,j\le 2$ we have$$(EA)_{ij} = \sum_{k=1}^2E_{ik}A_{kj} = E_{i1}A_{1j}+E_{i2}A_{2j}$$

If $i= 1$ then $$(EA)_{1j} = E_{11}A_{1j}+E_{12}A_{2j} = 0 \cdot A_{1j}+1\cdot A_{2j} = A_{2j}$$

If $i= 2$ then $$(EA)_{2j} = E_{21}A_{1j}+E_{22}A_{2j} = 1 \cdot A_{1j}+0\cdot A_{2j} = A_{1j}$$

So $$(EA)_{ij} = \begin{bmatrix}A_{21} & A_{22} \\ A_{11} & A_{12}\end{bmatrix}_{ij} = \begin{bmatrix}c & d \\ a & b\end{bmatrix}_{ij}$$