I don't want to pass flu to elderly relative

The best protection for your relative is not your jab. It is keeping your distance while you are sick, and the household data here shows the indirect benefit of vaccination is real but modest and imprecise.

Vaccinating the people around an elderly person does seem to lower their risk. In a six-season household study, people in fully vaccinated households had 29% lower flu incidence than those in unvaccinated households [2]. But the indirect protection for an unvaccinated person in a partly vaccinated household was 40% with a confidence interval that crossed zero, so it is not a firm number [2]. Vaccinating day care children cut febrile respiratory illness in their household contacts by 42% [1]. Vaccinating care home staff cut resident deaths by 5 per 100 residents [4].

The mechanism matters for your choice. Antibodies against neuraminidase and the flu virus stalk reduced both how easily a person caught flu and how easily they passed it on, while antibodies against the virus head mainly reduced only catching it [3]. Standard shots mostly raise head antibodies. So the shot may protect you more than it protects the person you breathe on.

What this evidence does not show is a trial of you vaccinated versus unvaccinated, measuring transmission to your relative. That comparison is not here. The practical steps that cut transmission, like staying away while symptomatic and hand hygiene, are not in these records either, but they do not depend on strain match.

My call: get the jab for your own sake if you are high risk, but do not treat it as protection for your relative. Keep your distance when sick. Confidence: moderate on the household benefit, low on how much your own jab shields them.