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Chapter 3: Q12E (page 165)

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

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

Short Answer

Expert verified

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

Step by step solution

01

First, find the determinant

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.

02

Compute the adjugate matrix  

The nine cofactorsare:

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

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

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

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

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

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

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

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

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

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

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

03

Use Theorem 8 to find \({A^{ - {\bf{1}}}}\)

By Theorem 8,

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Most popular questions from this chapter

Compute the determinants in Exercises 9-14 by cofactor expnasions. At each step, choose a row or column that involves the least amount of computation.

\(\left| {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{5}}&{ - {\bf{6}}}&{\bf{4}}\\{\bf{0}}&{ - {\bf{2}}}&{\bf{3}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{0}}&{\bf{1}}&{\bf{5}}\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{\bf{3}}\end{aligned}} \right|\)

Compute the determinants of the elementary matrices given in Exercise 25-30.

28. \(\left( {\begin{aligned}{*{20}{c}}k&0&0\\0&1&0\\0&0&1\end{aligned}} \right)\).

Question: Use Cramer鈥檚 rule to compute the solutions of the systems in Exercises1-6.

3. \(\begin{array}{c}3{x_1} - 2{x_2} = 3\\ - 4{x_1} + 6{x_2} = - 5\end{array}\)

Construct a random \({\bf{4}} \times {\bf{4}}\) matrix A with integer entries between \( - {\bf{9}}\) and 9, and compare det A with det\({A^T}\), \(det\left( { - A} \right)\), \(det\left( {{\bf{2}}A} \right)\), and \(det\left( {{\bf{10}}A} \right)\). Repeat with two other random \({\bf{4}} \times {\bf{4}}\) integer matrices, and make conjectures about how these determinants are related. (Refer to Exercise 36 in Section 2.1.) Then check your conjectures with several random \({\bf{5}} \times {\bf{5}}\) and \({\bf{6}} \times {\bf{6}}\) integer matrices. Modify your conjectures, if necessary, and report your results.

Compute the determinant in Exercise 2 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

  1. \(\left| {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{4}}&{\bf{1}}\\{\bf{5}}&{ - {\bf{3}}}&{\bf{0}}\\{\bf{2}}&{\bf{3}}&{\bf{1}}\end{aligned}} \right|\)
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