If $n$ (or $m$) is a multiple of $3$, then we can divide all lamps in each column (or row) into groups of $3$ and switch the lamps on by the groups.

If neither $n$ nor $m$ is a multiple of $3$, then color all lamps by diagonals with three colors (Fig. 10). Then each switching changes the state of exactly one lamp of each color, therefore each switching changes the parity of the number of lamps switched on for each color. Since in the beginning all lamps are off, the parity of the lamps switched on for each color always stays the same.

Let $n = 3a + b$ and $m = 3c + d$, where $a$ and $c$ are nonnegative integers and $b$ and $d$ are either $1$ or $2$. In regions of sizes $3a \times 3c$, $b \times 3c$ and $3a \times d$ the numbers of lamps of each color are equal and hence their parities are equal, but in the remaining $b \times d$ region there is either $1$ lamp (Fig. 11), $2$ lamps of different colors (Fig. 12 and 13), or $4$ lamps, of which $2$ are of one color and two others of the two other colors (Fig. 14), hence the parities of the numbers of lamps of each color are not equal. Consequently, it is not possible to switch on all lamps.

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14