<p>In this paper, we study a particular class of block matrices placing an emphasis on their spectral properties. Some related applications are then presented. In particular, we prove the existence of an infinite number of integer matrix solutions for the two equations <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_957_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">\(aX^{p}+bY^{p}=cZ^{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <msup> <mi>X</mi> <mi>p</mi> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>Y</mi> <mi>p</mi> </msup> <mo>=</mo> <mi>c</mi> <msup> <mi>Z</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_957_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="128" /> </InlineMediaObject> <EquationSource Format="TEX">\(aX^{p}+bY^{q}=cZ^{r}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <msup> <mi>X</mi> <mi>p</mi> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>Y</mi> <mi>q</mi> </msup> <mo>=</mo> <mi>c</mi> <msup> <mi>Z</mi> <mi>r</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for any integers <i>a</i>,&#xa0;<i>b</i> and <i>c</i> and for any positive integers <i>p</i>,&#xa0;<i>q</i> and <i>r</i> such that these solution matrices have all of their entries nonzero natural numbers.</p>