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The ideal strength of a material is the stress at which the lattice itself becomes unstable and, hence, sets a firm upper bound on the mechanical strength the material can have. The present paper includes an ab-initio calculation of the ideal shear strength of Fe. It is, to our knowledge, the first such computation for any ferromagnetic material. The paper also elaborates on our earlier calculation of the ideal tensile strength of Fe by studying the effects of strains which break the tetragonal symmetry. The strengths were calculated using the Projector Augmented Wave Method within the framework of density functional theory and the generalized gradient approximation. In <001> tension the ideal strength is determined by an elastic instability of the ferromagnetic phase along the "Bain" strain path from bcc to fcc. An <001> tensile strain also leads to instability with respect to transformation into a face centered orthorhombic structure, and to various magnetic instabilities. However, these are encountered at larger strains and, thus, do not affect the ideal strength. We also investigated the ideal shear strength of bcc iron in two prominent shear systems, <111>{112} and <111>{110}. In both shear systems the ideal strength is determined by the body centered tetragonal structure that defines a nearby saddle point on the energy surface. The ideal shear strengths are thus very similar, though they are not identical since the two shears follow slightly different strain paths from bcc to bct. We investigated the magnetic instabilities encountered during <111>{112} shear. These instabilities do not appear until the strain is significantly greater than the instability strain of the ferromagnetic crystal. Hence while Fe exhibits some novel effects due to magnetism, they do not affect the ideal strength, which is determined by the same elastic instabilities that determine the strengths of most other bcc metals.