Abstract In this manuscript, we study some algebraic and differential properties of matrix orthogonal polynomials with respect to the Laguerre-Sobolev right sesquilinear form defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msub> <m:mrow> <m:mrow> <m:mo>⟨</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>⟩</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi mathvariant="bold">S</m:mi> </m:mrow> </m:msub> <m:mo>≔</m:mo> <m:munderover> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msubsup> <m:mrow> <m:mi mathvariant="bold">W</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">L</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">A</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo>+</m:mo> <m:mi mathvariant="bold">M</m:mi> <m:munderover> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo accent="false">′</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mi mathvariant="bold">W</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>q</m:mi> <m:mo accent="false">′</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo>,</m:mo> </m:math> {\langle p,q\rangle }_{{\bf{S}}}:= \underset{0}{\overset{\infty }{\int }}{p}^{* }\left(x){{\bf{W}}}_{{\bf{L}}}^{{\bf{A}}}\left(x)q\left(x){\rm{d}}x+{\bf{M}}\underset{0}{\overset{\infty }{\int }}{(p^{\prime} \left(x))}^{* }{\bf{W}}\left(x)q^{\prime} \left(x){\rm{d}}x, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="bold">W</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">L</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">A</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>λ</m:mi> <m:mi>x</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="bold">A</m:mi> </m:mrow> </m:msup> </m:math> {{\bf{W}}}_{{\bf{L}}}^{{\bf{A}}}\left(x)={e}^{-\lambda x}{x}^{{\bf{A}}} is the Laguerre matrix weight, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="bold">W</m:mi> </m:math> {\bf{W}} is some matrix weight, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q are the matrix polynomials, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="bold">M</m:mi> </m:math> {\bf{M}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="bold">A</m:mi> </m:math> {\bf{A}} are the matrices such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="bold">M</m:mi> </m:math> {\bf{M}} is non-singular and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="bold">A</m:mi> </m:math> {\bf{A}} satisfies a spectral condition, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda is a complex number with positive real part.