Trouble with matrix multiplication in Julia

I'm having trouble with getting the same answer for the same matrix product in Julia.

I'm using Jupyter Lab and have written the following code:

using LinearAlgebra

function U1(α, M)
    return exp(α * M)
end

function U2(β, M)
    return exp(β * M)
end

test_list=[1 2]

α_list = test_list[1]
β_list = test_list[2]

M1=[1 3 3; 3 2 2; 2 2 1]
M2=[3 2 1; 3 0 3; 2 1 1]

init_vec=[1 ; 1 ; 1]

for _ in 1:2
    α=α_list[1]
    β=β_list[1]
    final_vec = U1(α,M1) * U2(β,M2) * init_vec
    println(final_vec)
end

#[3.2498540882389616e7, 3.362948386386128e7, 2.2549887512082998e7]
#[3.3668507705110244e7, 3.4777346320714206e7, 2.548964322110533e7]

I randomly set the values of test_list, init_vec, M1, and M2. As shown, I'm getting random-looking answers.

What could be causing this? How should I modify the code?

Running your code in the REPL I get:

[3.3668507705110244e7, 3.4777346320714206e7, 2.548964322110533e7]
[3.3668507705110244e7, 3.4777346320714206e7, 2.548964322110533e7]

When I replace test_list, M1, M2, and init_vec with appropriately sized random inputs I get:

[1.6282115006640232, 0.9234175545808518, 3.517098945783593]
[1.6282115006640232, 0.9234175545808518, 3.517098945783593]

In other words, I can't reproduce the behaviour you are observing. Have you tried to run the exact code you posted in a fresh Julia session?

Edited to add: Based on the comments I also ran your code in an IJulia Jupyter notebook and am getting:

Note that using LinearAlgebra is not required here, you are not actually using any functions that aren't in Base.

As Oscar says, it's not a good idea to install Julia from sources other than the official binaries, so maybe try that.