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For which value of $a$ in the matrix {{a,2}, {3,4}} has the matrix eigenvalue of $0$? How to solve it in Mathematica?

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closed as off-topic by Dr. belisarius, user9660, MarcoB, Edmund, WReach Feb 6 '16 at 3:40

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    $\begingroup$ Do you need to do this by hand or using Mathematica? $\endgroup$ – Coolwater Feb 5 '16 at 12:01
  • $\begingroup$ I have to use mathematica. $\endgroup$ – ASPire Feb 5 '16 at 12:03
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    $\begingroup$ Then look up the function Eigenvalues in the documentation $\endgroup$ – Coolwater Feb 5 '16 at 12:04
  • $\begingroup$ I found it how to find, but not how to find for the value a. :( $\endgroup$ – ASPire Feb 5 '16 at 12:05
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    $\begingroup$ People have voted your question down very probably because you didn't show due diligence. It looks like you didn't bother to search on the documentation (or showed us that you did). You didn't even bother to write the matrix in the proper syntax. Only good questions are likely to get great answers. $\endgroup$ – rhermans Feb 5 '16 at 13:21
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Another way to go is using NMinimize

NMinimize[Min@Abs@Eigenvalues[{{a, 2}, {3, 4}}], a]
(* {3.89083*10^-10, {a -> 1.5}} *)

which of course matches the analytic answer above

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To get the expresions for the Eigenvalues

Eigenvalues[{{a, 2} , {3, 4}}]
{1/2 (4 + a - Sqrt[40 - 8 a + a^2]),  1/2 (4 + a + Sqrt[40 - 8 a + a^2])}

make each expresion Equal to zero (f@x is the Prefix form of f[x])

 Thread@Equal[Eigenvalues[{{a, 2} , {3, 4}}], 0]
{1/2 (4 + a - Sqrt[40 - 8 a + a^2]) == 0, 1/2 (4 + a + Sqrt[40 - 8 a + a^2]) == 0}

then Solve each (/@ is a shortcut for Map)

Solve /@ Thread@Equal[Eigenvalues[{{a, 2} , {3, 4}}], 0]
{{{a -> 3/2}}, {}}
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