<p>Let the Jordan canonical form of a nonsingular <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(3 \times 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> matrix <i>A</i> be <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(J=J_3[\lambda ]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo>=</mo> <msub> <mi>J</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">[</mo> <mi>λ</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\lambda \ne 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We find all non-trivial solutions of the Yang–Baxter-like matrix equation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(JYJ=YJY\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mi>Y</mi> <mi>J</mi> <mo>=</mo> <mi>Y</mi> <mi>J</mi> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> in this paper. Based on this result, we can use the similarity transformation and obtain all non-trivial solutions of Yang–Baxter-like matrix equation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(AXA=XAX\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>A</mi> <mo>=</mo> <mi>X</mi> <mi>A</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>. We also investigate the intrinsic structure of the non-trivial solution set and prove that it is path-connected. Additionally, using our approach, we can find solutions to the Yang–Baxter-like equation with a single parameter.</p>