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Question

Find the inverse of matrix by adjoint method
$$A=\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{matrix} \right] $$


Solution

$$A=\left( \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{matrix} \right) $$
$$|A|=1(6-1)-2(4-3)+3 (2-6)$$
$$5-2-12$$
$$-9$$
Minor of A$$=\left( \begin{matrix} 5 & 1 & -4 \\ 1 & -4 & -5 \\ -4 & -5 & -1 \end{matrix} \right) $$
$$C of (A)=\left( \begin{matrix} 5 & -1 & -4 \\ -1 & -4 & +5 \\ -4 & +5 & -1 \end{matrix} \right) $$
adj (A)$$=\left( \begin{matrix} 5 & -1 & -4 \\ -1 & -1 & 5 \\ -4 & 5 & -1 \end{matrix} \right) $$
$$\displaystyle A^{-1}=\frac { adj (A)}{ |A| } = \frac {1 }{ 9 } \left( \begin{matrix} -5 & 1 & 4 \\ 1 & 4 & -5 \\ 4 & -5 & 1 \end{matrix} \right) $$
$$=\frac {1 }{ 9 } \left( \begin{matrix} -5 & 1 & 4 \\ -1 & 4 & -5 \\ 4 & -5 & 1 \end{matrix} \right) $$

Mathematics

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