Question Help

No it will not.
You don't need a mixer if the PC is your only source. That mixer doesn't have an amp built into it. Some do so if you want to use multiple sources and blend them together you will need a mixer-amp.
If not you don't need a mixer at all.
To use passive loudspeakers you will need an amplifier. You need either an integrated amp (that will have more than one input and a volume control) or a basic power amp (one input and sometimes a volume control).
Those speakers are rated for 100 watts into 8 ohms so you would want about that amount of power. Power ratings on some low priced amps are very deceptive or outright lies so you need to check the specs carefully. The ones I link to have highly exaggerated power specs at the top of the page but it you dig down you will set that the mixer is 100w per channel into 8 ohms and the basic amp 125w per channel into 8 ohms. Either would be fine for the speakers.