Find the inverse of the matrix (if it exists) given \(\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos \alpha }&{\sin \alpha } \\ 0&{\sin \alpha }&{ - \cos \alpha } \end{array}} \right]\)
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Find the inverse of the matrix (if it exists) given \(\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos \alpha }&{\sin \alpha } \\ 0&{\sin \alpha }&{ - \cos \alpha } \end{array}} \right]\)
Answer
Let \(A = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos \alpha }&{\sin \alpha } \\ 0&{\sin \alpha }&{ - \cos \alpha } \end{array}} \right]\)
\(\therefore \left| A \right| = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos \alpha }&{\sin \alpha } \\ 0&{\sin \alpha }&{ - \cos \alpha } \end{array}} \right]\)
\( = \left( { - {{\cos }^2}\alpha - {{\sin }^2}\alpha } \right) - 0 + 0 \) \(= - \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right) = - 1 \ne 0\)
\(\therefore {A^{ - 1}}\) exists.
\({A_{11}} = + \left| {\begin{array}{*{20}{c}} {\cos \alpha }&{\sin \alpha } \\ {\sin \alpha }&{ - \cos \alpha } \end{array}} \right| = + \left( { - {{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\)
\(= - \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right) = - 1\)
\({A_{12}} = - \left| {\begin{array}{*{20}{c}} 0&{\sin \alpha } \\ 0&{ - \cos \alpha } \end{array}} \right| = - \left( {0 - 0} \right) = 0,\) \({A_{13}} = + \left| {\begin{array}{*{20}{c}} 0&{\cos \alpha } \\ 0&{\sin \alpha } \end{array}} \right| = + \left( {0 - 0} \right) = 0\)
\({A_{21}} = - \left| {\begin{array}{*{20}{c}} 0&0 \\ {\sin \alpha }&{-\cos \alpha } \end{array}} \right| = - \left( {0 - 0} \right) = 0,\) \({A_{22}} = + \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{-\cos \alpha } \end{array}} \right| = + \left( { - \cos \alpha - 0} \right) = - \cos \alpha \)
\({A_{23}} = - \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{\sin \alpha } \end{array}} \right| = - \left( {\sin \alpha - 0} \right) = \sin \alpha ,\) \({A_{31}} = + \left| {\begin{array}{*{20}{c}} 0&0 \\ {\cos \alpha }&{\sin \alpha } \end{array}} \right| = \left( {0 - 0} \right) = 0\)
\({A_{32}} = - \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{\sin \alpha } \end{array}} \right| = - \left( {\sin \alpha - 0} \right) = - \sin \alpha \), \({A_{33}} = + \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{\cos \alpha } \end{array}} \right| = + \left( {\cos \alpha - 0} \right) = \cos \alpha \)
\(\therefore adj.A = \left[ {\begin{array}{*{20}{c}} { - 1}&0&0 \\ 0&{ - \cos \alpha }&{ \sin \alpha } \\ 0&{ - \sin \alpha }&{\cos \alpha } \end{array}} \right]' \) \(= \left[ {\begin{array}{*{20}{c}} { - 1}&0&0 \\ 0&{ - \cos \alpha }&{ -\sin \alpha } \\ 0&{ \sin \alpha }&{\cos \alpha } \end{array}} \right]\)
\(\therefore {A^{ - 1}} = \frac{1}{{\left| A \right|}}adj.A \) \(= - \left[ {\begin{array}{*{20}{c}} { - 1}&0&0 \\ 0&{ - \cos \alpha }&{ - \sin \alpha } \\ 0&{ \sin \alpha }&{\cos \alpha } \end{array}} \right] \) \(= \left[ {\begin{array}{*{20}{c}} { 1}&0&0 \\ 0&{ \cos \alpha }&{\sin \alpha } \\ 0&{ - \sin \alpha }&{ - \cos \alpha } \end{array}} \right]\)
\(\therefore \left| A \right| = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{\cos \alpha }&{\sin \alpha } \\ 0&{\sin \alpha }&{ - \cos \alpha } \end{array}} \right]\)
\( = \left( { - {{\cos }^2}\alpha - {{\sin }^2}\alpha } \right) - 0 + 0 \) \(= - \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right) = - 1 \ne 0\)
\(\therefore {A^{ - 1}}\) exists.
\({A_{11}} = + \left| {\begin{array}{*{20}{c}} {\cos \alpha }&{\sin \alpha } \\ {\sin \alpha }&{ - \cos \alpha } \end{array}} \right| = + \left( { - {{\cos }^2}\alpha - {{\sin }^2}\alpha } \right)\)
\(= - \left( {{{\cos }^2}\alpha + {{\sin }^2}\alpha } \right) = - 1\)
\({A_{12}} = - \left| {\begin{array}{*{20}{c}} 0&{\sin \alpha } \\ 0&{ - \cos \alpha } \end{array}} \right| = - \left( {0 - 0} \right) = 0,\) \({A_{13}} = + \left| {\begin{array}{*{20}{c}} 0&{\cos \alpha } \\ 0&{\sin \alpha } \end{array}} \right| = + \left( {0 - 0} \right) = 0\)
\({A_{21}} = - \left| {\begin{array}{*{20}{c}} 0&0 \\ {\sin \alpha }&{-\cos \alpha } \end{array}} \right| = - \left( {0 - 0} \right) = 0,\) \({A_{22}} = + \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{-\cos \alpha } \end{array}} \right| = + \left( { - \cos \alpha - 0} \right) = - \cos \alpha \)
\({A_{23}} = - \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{\sin \alpha } \end{array}} \right| = - \left( {\sin \alpha - 0} \right) = \sin \alpha ,\) \({A_{31}} = + \left| {\begin{array}{*{20}{c}} 0&0 \\ {\cos \alpha }&{\sin \alpha } \end{array}} \right| = \left( {0 - 0} \right) = 0\)
\({A_{32}} = - \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{\sin \alpha } \end{array}} \right| = - \left( {\sin \alpha - 0} \right) = - \sin \alpha \), \({A_{33}} = + \left| {\begin{array}{*{20}{c}} 1&0 \\ 0&{\cos \alpha } \end{array}} \right| = + \left( {\cos \alpha - 0} \right) = \cos \alpha \)
\(\therefore adj.A = \left[ {\begin{array}{*{20}{c}} { - 1}&0&0 \\ 0&{ - \cos \alpha }&{ \sin \alpha } \\ 0&{ - \sin \alpha }&{\cos \alpha } \end{array}} \right]' \) \(= \left[ {\begin{array}{*{20}{c}} { - 1}&0&0 \\ 0&{ - \cos \alpha }&{ -\sin \alpha } \\ 0&{ \sin \alpha }&{\cos \alpha } \end{array}} \right]\)
\(\therefore {A^{ - 1}} = \frac{1}{{\left| A \right|}}adj.A \) \(= - \left[ {\begin{array}{*{20}{c}} { - 1}&0&0 \\ 0&{ - \cos \alpha }&{ - \sin \alpha } \\ 0&{ \sin \alpha }&{\cos \alpha } \end{array}} \right] \) \(= \left[ {\begin{array}{*{20}{c}} { 1}&0&0 \\ 0&{ \cos \alpha }&{\sin \alpha } \\ 0&{ - \sin \alpha }&{ - \cos \alpha } \end{array}} \right]\)
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