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We study the electronic and magnetic properties of T$_2$AlC (T=Ti and Cr) compounds in the density-functional theory using the generalized gradient approximation (GGA) with consideration of strong Coulomb correlations (GGA+$U$) in the framework of the fully relativistic spin-polarized Dirac linear muffin-tin orbital (LMTO) band-structure method. The X-ray absorption spectra and X-ray magnetic circular dichroism (XMCD) at the Cr $L_{2,3}$ and Cr, Ti, and C $K$ edges were investigated theoretically. The calculated results are in good agreement with experimental data. The effect of the electric quadrupole $E_2$ and magnetic dipole $M_1$ transitions at the Cr $K$ edge has been investigated., Comment: 16 pages, 7 figures, 1 table. arXiv admin note: text overlap with arXiv:cond-mat/0210482 by other authors