Let A=[2103] be a matrix such that A10=[abcd] , then the number of positive integral factors of a+b+c+d is ‘n’ where (n−15) is

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Let $A = [\begin{matrix} 2 & 1 \\ 0 & 3 \end{matrix}]$ be a matrix such that $A^{10} = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then the number of positive integral factors of $a + b + c + d$ is ‘n’ where $(n - 15)$ is

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Detailed Solution

$A^{2} = [\begin{array}{l} \begin{matrix} 2^{2} & 3^{2} & - 2^{2} \end{matrix} \\ \begin{matrix} 0 & 3^{2} \end{matrix} \end{array}], A^{10} = [\begin{array}{l} \begin{matrix} 2^{10} & 3^{10} & - 2^{10} \end{matrix} \\ \begin{matrix} 0 & 3^{10} \end{matrix} \end{array}] ∴ a + b + c + d = 2^{1} . 3^{10}$
$∴$ Number of factors $= 2 \times 11 = 22$