Let $A = (\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix})$ ; Trace (A) = 9; a, b, c are positive integers such that ab + bc + ca = 26. Let $A_{1}$ denote adjoint of A, $A_{2}$ denote the adjoint of $A_{1}$ ……..and so on. If the value of $| A_{4} |$ is M then which of the following statement(s) is(are) CORRECT.

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last two digits of M are 6 and 1

last three digits of M are 8, 6 and 1

$M = 3^{24}$

$M = 3^{48}$

answer is A, C.

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Detailed Solution

$a + b + c = 9, a b + b c + c a = 26$

$| A | = 3 a b c - a^{3} - b^{3} - c^{3} = - 27$

$| A_{1} | = {| A |}^{2} ∴ | A_{4} | = 3^{48} = M$

Last three digits are 3, 6, 1

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