Will the black hole at the center of the Milky Way eventually eat all the matter in our galaxy?

If the universe is destined to expand forever, the answer to this is believed to be yes, but the time for this to happen is...astronomical.

The galaxy will be eaten from the inside out as the orbits of the stars in the galaxy continue to lose energy via the emission of gravitational radiation. The amount of orbital energy a star near the sun's distance loses is incredibly small, but over the course of an estimated 10^60 years (that's ten to the 60th power) or so, enough energy will be lost that most stars will have sunk into the center of our galaxy and be gobbled up by the 100 billion solar mass black hole that has now formed there by eating up all the other stars interior to the Sun's orbit around the galactic center. After something line 10^160 years or so, even this black hole will vanish because, as all black holes are believed to do, it too is radiating energy and evaporating. In the end, all that is left over in the universe will be no black holes and a thinning soup of electrons, neutrinos, anti-electrons and photons.