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Adjoint of a Matrix

If A=[ 1 2 ...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 1 3 & 2 & 3 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$ , then find $A^{- 1}$ .

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Solution

$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 1 3 & 2 & 3 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$

$A^{- 1} = \frac{a d j A}{| A |}$

$A^{- 1} = \frac{⎡ ⎢ ⎣ \begin{matrix} (2 \times 2 - 3 \times 1) & - (2 \times 2 - 1 \times 1) & (2 \times 3 - 2 \times 1) - (3 \times 2 - 3 \times 1) & (1 \times 2 - 1 \times 1) & - (1 \times 3 - 3 \times 1) (3 \times 3 - 2 \times 1) & - (1 \times 1 - 2 \times 1) & (1 \times 2 - 2 \times 3) \end{matrix} ⎤ ⎥ ⎦}{| 1 (2 \times 2 - 3 \times 1) - 3 (2 \times 2 - 1 \times 1) + 1 (2 \times 3 - 2 \times 1) |}$

$A^{- 1} = \frac{⎡ ⎢ ⎣ \begin{matrix} 1 & - 3 & 4 - 3 & 1 & 0 1 & 1 & - 4 \end{matrix} ⎤ ⎥ ⎦}{1 - 9 + 4}$

$A^{- 1} = \frac{⎡ ⎢ ⎣ \begin{matrix} 1 & - 3 & 4 - 3 & 1 & 0 1 & 1 & - 4 \end{matrix} ⎤ ⎥ ⎦}{- 4}$

$A^{- 1} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{- 1}{4} & \frac{3}{4} & - 1 \frac{3}{4} & \frac{- 1}{4} & 0 \frac{- 1}{4} & \frac{- 1}{4} & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

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