We aren’t experts as such here, but your photo quality isn’t good enough to tell much at all.
I suggest you Repost your question in “Colored gems” and include a daylight / outside photo as well as one inside using a torchlight (for incandescent light).
It should be a blue / green colour in daylight and change to a purple / red colour when a torch light is used if it is an Alexandrite.
There are other gems that show a colour change and plenty of types of synthetic Alexandrite around, both modern and vintage/ antique.
Yes. The GIA. Unfortunately you cannot tell from a picture.
They can manufacture Alexandrite--it's real alexandrite, chemically, physically, optically identical. It's impossible to tell from a picture. In fact, in one case GIA famously labeled a synthetic as real:
Most "Alexandrite" isn't even synthetic Alexandrite. It's color-change sapphire. They sell it in Alexandria,
Egypt, even though the Alexandra in question was a Russian Czarina. The sapphires are easier to spot, as they range in color from bluish-purple to reddish purple. The majority of people who have grandma's Alexandrite ring and want to know what it's worth will turn out to have a color change sapphire.
Alexandrites are extremely expensive. Often that alone is enough to answer the question. There are moderately priced stones, but they will be very included and/or the color change will be minimal, from a muddy green to a muddy reddish-brown.
There is also some controversy over what gets to be Alexandrite and what is downgraded to color-change chyrsobel. Sticklers argue that only stones from the original mines in Russia count, and even that only the very best color changes count. Those mines were long ago depleted and almost all stones on the market now are from Africa. There are also stones being represented as from a secret mine in Russia that was not depleted, smuggled out of the Soviet Union/Russia, etc. Which, they're not. Sigh.
This wouldn't be possible with even the best of photos. Fortunately, it's not usually that difficult in person with the right tools and a bit of practice. There's a list of local appraisers under the 'resources' tab at the top of the page who should be able to help you out.