2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58905As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras over the real number field ${\cal R}$, because ${\cal O}$ is a non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an extension of ${\cal H}$ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.23 pages, LaTexRings and Algebras15A33; 15A06; 15A24; 17A35text