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Finding Inverse Using Elementary Transformations

Find the inve...

Question

Find the inverse of $A = ⎡ ⎢ ⎣ \begin{matrix} cos α & - sin α & 0 sin α & cos α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$ using elementary transformation.

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Solution

$⎡ ⎢ ⎣ \begin{matrix} c o s α & - sin α & 0 sin α & cos α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$

$R_{1} \to R_{1} cos α R_{2} \to R_{2} sin α$

$⎡ ⎢ ⎣ \begin{matrix} {cos}^{2} α & - cos α sin α & 0 {sin}^{2} α & cos α sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} cos α & 0 & 0 0 & sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$

$R_{1} \to R_{1} + R_{2}$

$⎡ ⎢ ⎣ \begin{matrix} {cos}^{2} α + {sin}^{2} α & - cos α sin α + cos α sin α & 0 {sin}^{2} α & cos α sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} cos α & sin α & 0 0 & sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$

$⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 {sin}^{2} α & cos α sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} cos α & sin α & 0 0 & sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$

$R_{2} \to R_{2} - R_{1} {sin}^{2} α$

$⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 {sin}^{2} α - {sin}^{2} α & cos α sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} cos α & sin α & 0 - cos α {sin}^{2} α & sin α - {sin}^{3} α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$

$⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & cos α sin α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} cos α & sin α & 0 - cos α {sin}^{2} α & sin α {cos}^{2} α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$

$R_{2} \to \frac{R_{2}}{cos α sin α}$

$⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} cos α & sin α & 0 - sin α & cos α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ A$

$A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} cos α & sin α & 0 - sin α & cos α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$

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