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Symmetric Matrix

Let ABC = I t...

Question

Let $A B C = I$ then $tr (A B C + B C A + C A B)$ is
(where order of matrices $A, B, C$ is $3$ and $tr (A)$ is sum of the principle diagonal elements in $A$ )

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Solution

$A B C = I$
$B C = A^{- 1}$
$B C A = I$

$B C = A^{- 1}$
$C = B^{- 1} A^{- 1}$
$C A = B^{- 1} \Rightarrow C A B = I$

$tr (3 I) = 9$

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A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦

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| A | =

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