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Let <i>G</i> be a simple undirected graph on <i>n</i> vertices. V. Nikiforov studied hybrids of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>G</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>G</mi></msub></semantics></math></inline-formula> and defined the matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>A</mi><mrow><mi>α</mi></mrow><mi>G</mi></msubsup></semantics></math></inline-formula> for every real <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mrow><mi>α</mi></mrow><mi>G</mi></msubsup><mo>=</mo><mi>α</mi><msub><mi>D</mi><mi>G</mi></msub><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><msub><mi>A</mi><mi>G</mi></msub><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, we define the generalized Randić matrix for graph <i>G</i>, and we introduce and establish bounds for the Estrada index of this new matrix. Furthermore, we find the smallest value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> for which the generalized Randić matrix is positive semidefinite. Finally, we present the solution to the problem proposed by V. Nikiforov. The problem consists of the following: for a given simple undirected graph <i>G</i>, determine the smallest value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>A</mi><mrow><mi>α</mi></mrow><mi>G</mi></msubsup></semantics></math></inline-formula> is positive semidefinite.