The aim of this paper is to study representations of 3-dimensional simple multiplicative Hom-Lie algebras <svg style="vertical-align:-2.85284pt;width:69.637497px;" id="M1" height="15.75" version="1.1" viewBox="0 0 69.637497 15.75" width="69.637497" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path id="x1D524" d="M414 318v-206q0 -50 23 -94q24 -44 24 -72q0 -72 -62 -113q-61 -42 -136 -42q-45 0 -72 10q-28 9 -38 17q-3 3 -21 25q-13 19 -27 26.5t-48 9.5v20q60 25 104 25q52 0 72 -62q5 -17 16.5 -29.5t21.5 -12.5q47 0 69 19.5t22 67.5q0 27 -23 70q-21 40 -26 84h-1
q-73 -45 -157 -79h-14q-90 75 -90 209q0 49 18 93t51 71q17 15 79 55q61 40 80 58h14l32 -22l21.5 -13.5l18.5 -11.5q47 -27 90 -39v-14q-19 -1 -30 -15t-11 -35zM312 97v239l-92 53q-67 -67 -67 -190q0 -57 18 -101q19 -42 50 -42q14 0 48 14q34 15 43 27z" /></g><g transform="matrix(.017,-0,0,-.017,14.426,12.138)"><path id="x3B" d="M114 412q23 0 39 -16t16 -41t-16 -41.5t-40 -16.5t-39.5 16.5t-15.5 41.5t15.5 41t40.5 16zM95 130q31 0 61 -30t30 -78q0 -53 -38 -88t-92 -52l-11 30q76 32 76 85q0 26 -17.5 43.5t-44.5 23.5q-13 3 -13 24q0 19 14.5 30.5t34.5 11.5z" /></g><g transform="matrix(.017,-0,0,-.017,21.124,12.138)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,26.989,12.138)"><path id="x22C5" d="M170 255q0 -25 -16 -41t-40 -16t-40 16t-16 41t16 41.5t40 16.5t40 -16.5t16 -41.5z" /></g><g transform="matrix(.017,-0,0,-.017,30.864,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,37.579,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.454,12.138)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g><g transform="matrix(.017,-0,0,-.017,47.319,12.138)"><use xlink:href="#x3B"/></g><g transform="matrix(.017,-0,0,-.017,54.017,12.138)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204
q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g><g transform="matrix(.017,-0,0,-.017,63.672,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> (whose structure is of <i>A</i><sub>1</sub>-type). In this paper we can see that a finite dimensional representation of <svg style="vertical-align:-2.85284pt;width:69.637497px;" id="M2" height="15.75" version="1.1" viewBox="0 0 69.637497 15.75" width="69.637497" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x1D524"/></g><g transform="matrix(.017,-0,0,-.017,14.426,12.138)"><use xlink:href="#x3B"/></g><g transform="matrix(.017,-0,0,-.017,21.124,12.138)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,26.989,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.864,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.579,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.454,12.138)"><use xlink:href="#x5D"/></g><g transform="matrix(.017,-0,0,-.017,47.319,12.138)"><use xlink:href="#x3B"/></g><g transform="matrix(.017,-0,0,-.017,54.017,12.138)"><use xlink:href="#x1D6FC"/></g><g transform="matrix(.017,-0,0,-.017,63.672,12.138)"><use xlink:href="#x29"/></g> </svg> is not always completely reducible, and a representation of <svg style="vertical-align:-2.85284pt;width:69.637497px;" id="M3" height="15.75" version="1.1" viewBox="0 0 69.637497 15.75" width="69.637497" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x1D524"/></g><g transform="matrix(.017,-0,0,-.017,14.426,12.138)"><use xlink:href="#x3B"/></g><g transform="matrix(.017,-0,0,-.017,21.124,12.138)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,26.989,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.864,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.579,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.454,12.138)"><use xlink:href="#x5D"/></g><g transform="matrix(.017,-0,0,-.017,47.319,12.138)"><use xlink:href="#x3B"/></g><g transform="matrix(.017,-0,0,-.017,54.017,12.138)"><use xlink:href="#x1D6FC"/></g><g transform="matrix(.017,-0,0,-.017,63.672,12.138)"><use xlink:href="#x29"/></g> </svg> is irreducible if and only if it is a regular Lie-type representation.