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Q12E

Expert-verifiedFound in: Page 165

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**Question: In Exercise 12, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.**

**12. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{1}}&{\bf{3}}\\{ - {\bf{2}}}&{\bf{2}}&{\bf{1}}\\{\bf{0}}&{\bf{1}}&{\bf{1}}\end{array}} \right)\)**

The adjugate matrix is \(\left( {\begin{array}{*{20}{c}}1&2&{ - 5}\\2&1&{ - 7}\\{ - 2}&{ - 1}&4\end{array}} \right)\), and the inverse matrix is

Let \(A = \left( {\begin{array}{*{20}{c}}1&1&3\\{ - 2}&2&1\\0&1&1\end{array}} \right)\). Then,

\(\begin{array}{c}\det A = \left| {\begin{array}{*{20}{c}}1&1&3\\{ - 2}&2&1\\0&1&1\end{array}} \right|\\ = 0 - 1\left| {\begin{array}{*{20}{c}}1&3\\{ - 2}&1\end{array}} \right| + 1\left| {\begin{array}{*{20}{c}}1&1\\{ - 2}&2\end{array}} \right|\\ = - \left( 7 \right) + 4\\\det A = - 3 \ne 0\end{array}\)

Here, \(\det A \ne 0\). Hence, the inverse of *A* exists.

The nine **cofactors** are:

\(\begin{array}{c}{C_{11}} = {\left( { - 1} \right)^2}\left| {\begin{array}{*{20}{c}}2&1\\1&1\end{array}} \right|\\ = 1\end{array}\)

\(\begin{array}{c}{C_{12}} = {\left( { - 1} \right)^3}\left| {\begin{array}{*{20}{c}}{ - 2}&1\\0&1\end{array}} \right|\\ = - \left( { - 2} \right)\\ = 2\end{array}\)

\(\begin{array}{c}{C_{13}} = {\left( { - 1} \right)^4}\left| {\begin{array}{*{20}{c}}{ - 2}&2\\0&1\end{array}} \right|\\ = - 2\end{array}\)

\(\begin{array}{c}{C_{21}} = {\left( { - 1} \right)^3}\left| {\begin{array}{*{20}{c}}1&3\\1&1\end{array}} \right|\\ = - \left( { - 2} \right)\\ = 2\end{array}\)

\(\begin{array}{c}{C_{22}} = {\left( { - 1} \right)^4}\left| {\begin{array}{*{20}{c}}1&3\\0&1\end{array}} \right|\\ = 1\end{array}\)

\(\begin{array}{c}{C_{23}} = {\left( { - 1} \right)^5}\left| {\begin{array}{*{20}{c}}1&1\\0&1\end{array}} \right|\\ = - 1\end{array}\)

\(\begin{array}{c}{C_{31}} = {\left( { - 1} \right)^4}\left| {\begin{array}{*{20}{c}}1&3\\2&1\end{array}} \right|\\ = - 5\end{array}\)

\(\begin{array}{c}{C_{32}} = {\left( { - 1} \right)^5}\left| {\begin{array}{*{20}{c}}1&3\\{ - 2}&1\end{array}} \right|\\ = - 7\end{array}\)

\(\begin{array}{c}{C_{33}} = {\left( { - 1} \right)^6}\left| {\begin{array}{*{20}{c}}1&1\\{ - 2}&2\end{array}} \right|\\ = 4\end{array}\)

The **adjugate matrix is the transpose of the matrix of cofactors**. Hence,

\(\begin{array}{c}{\rm{adj}}\,A = \left( {\begin{array}{*{20}{c}}{{C_{11}}}&{{C_{21}}}&{{C_{31}}}\\{{C_{12}}}&{{C_{22}}}&{{C_{32}}}\\{{C_{13}}}&{{C_{23}}}&{{C_{33}}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}1&2&{ - 5}\\2&1&{ - 7}\\{ - 2}&{ - 1}&4\end{array}} \right)\end{array}\)

By **Theorem 8**,

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