Let A be a $3 \times 3$ matrix such that $| \text{adj (adj A)}| = 81$. If $S=\{ n \in Z : |(\text{adj (adj A)|}^{\frac{(n-1)^2}{2}} = |A|^{(3n^2-5n-4)} \}$, then $\sum_{n \in S} |A|^{(n^2+n)}$ is equal to
732
\(|\operatorname{adj}(\operatorname{adj})(\operatorname{adj} A)|=81\)
\(\Rightarrow|\operatorname{adj} \mathrm{A}|^{4}=81\)
\(\Rightarrow|\operatorname{adj} A|=3\)
\(\Rightarrow|A|^{2}=3\)
\(\Rightarrow|\mathrm{A}|=\sqrt{3}\)
\(\left(|A|^{4}\right)^{\frac{(\mathrm{n}-1)^{2}}{2}}=|\mathrm{A}|^{3 n^{2}-5 n-4}\)
\(\Rightarrow 2(\mathrm{n}-1)^{2}=3 \mathrm{n}^{2}-5 \mathrm{n}-4\)
\(\Rightarrow 2 \mathrm{n}^{2}-4 \mathrm{n}+2=3 \mathrm{n}^{2}-5 \mathrm{n}-4\)
\(\Rightarrow \mathrm{n}^{2}-\mathrm{n}-6=0\)
\(\Rightarrow(\mathrm{n}-3)(\mathrm{n}+2)=0\)
\(\Rightarrow \mathrm{n}=3,-2\)
\(\sum\limits_{\mathrm{n} \in \mathrm{S}}\left|\mathrm{A}^{\mathrm{n}^{2}+\mathrm{n}}\right|\)
\(=\left|\mathrm{A}^{2}\right|+\left|\mathrm{A}^{12}\right|\)
= 3 + 36 = 3 + 729 = 732
Similar Questions
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Important Questions from Algebra
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- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
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