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Question

Show that the determinant 
$$\left| {\begin{array}{*{20}{c}}  {{a^2} + {b^2} + {c^2}}&{bc + ca + ab}&{bc + ca + ab} \\   {bc + ca + ab}&{{a^2} + {b^2} + {c^2}}&{bc + ca + ab} \\   {bc + ca + ab}&{bc + ca + ab}&{{a^2} + {b^2} + {c^2}} \end{array}} \right|$$
is always non-negative.


Solution

$$\left| \begin{matrix} { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } & bc+ca+ab & bc+ca+ab \\ bc+ca+ab & { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } & bc+ca+ab \\ bc+ca+ab & bc+ca+ab & { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \end{matrix} \right| \\ =\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right| \left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right| \\ ={ \left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right|  }^{ 2 }\\ ={ (a+b+c) }^{ 2 }{ ({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }-ab-bc-ca) }^{ 2 }>0$$
for all $$a,b,c,a\neq b\neq c$$
$$\therefore$$ it always non-negative.(positive).

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