Chapter 3: Q30E (page 165)
Use Theorem 3 (but not Theorem 4) to show that if two rows of a square matrix A are equal, then \(det A = 0\). The same is true for twocolumns. Why?
It is proved thatif two rows or columns of asquare matrixA are equal, then \(\det A = 0\).
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Find the determinant in Exercise 16, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].
16. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{{\bf{5d}}}&{{\bf{5e}}}&{{\bf{5f}}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right|\]
Question: In Exercise 14, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.
14. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{1}}}&{\bf{2}}\\{\bf{0}}&{\bf{2}}&{\bf{1}}\\{\bf{2}}&{\bf{0}}&{\bf{4}}\end{array}} \right)\)
The expansion of a \({\bf{3}} \times {\bf{3}}\) determinant can be remembered by the following device. Write a second type of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals.
Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises 15-18. Warning: This trick does not generalize in any reasonable way to \({\bf{4}} \times {\bf{4}}\) or larger matrices.
\(\left| {\begin{aligned}{*{20}{c}}{\bf{2}}&{ - {\bf{3}}}&{\bf{3}}\\{\bf{3}}&{\bf{2}}&{\bf{2}}\\{\bf{1}}&{\bf{3}}&{ - {\bf{1}}}\end{aligned}} \right|\)
Question: Use Cramer’s rule to compute the solutions of the systems in Exercises1-6.
2. \(\begin{array}{l}4{x_1} + {x_2} = 6\\3{x_1} + 2{x_2} = 7\end{array}\)
Each equation in Exercises 1-4 illustrates a property of determinants. State the property
\(\left| {\begin{array}{*{20}{c}}{\bf{0}}&{\bf{5}}&{ - {\bf{2}}}\\{\bf{1}}&{ - {\bf{3}}}&{\bf{6}}\\{\bf{4}}&{ - {\bf{1}}}&{\bf{8}}\end{array}} \right| = - \left| {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{3}}}&{\bf{6}}\\{\bf{0}}&{\bf{5}}&{ - {\bf{2}}}\\{\bf{4}}&{ - {\bf{1}}}&{\bf{8}}\end{array}} \right|\)
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