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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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In Exercises 27 and 28, A and B are \[n \times n\] matrices. Mark each statement True or False. Justify each answer.
27. a. A row replacement operation does not affect the determinant of a matrix.
b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by \({\left( { - {\bf{1}}} \right)^r}\), where r is the number of row interchanges made during row reduction from A to U.
c. If the columns of A are linearly dependent, then \(det\left( A \right) = 0\).
d. \(det\left( {A + B} \right) = det{\rm{ }}A + det{\rm{ }}B\).
Question: 11. Find the area of the parallelogram determined by the points \(\left( {1,4} \right),\)\(\left( { - 1,5} \right),\)\(\left( {3,9} \right),\) and \(\left( {5,8} \right)\). How can you tell that the quadrilateral determined by the points is actually a parallelogram?
Compute the determinant in Exercise 10 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.
10. \(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{2}}}&{\bf{5}}&{\bf{2}}\\{\bf{0}}&{\bf{0}}&{\bf{3}}&{\bf{0}}\\{\bf{2}}&{ - {\bf{4}}}&{ - {\bf{3}}}&{\bf{5}}\\{\bf{2}}&{\bf{0}}&{\bf{3}}&{\bf{5}}\end{array}} \right|\)
Question: In Exercises 31鈥�36, mention an appropriate theorem in your explanation.
34. Let A and P be square matrices, with P invertible. Show that \(det\left( {PA{P^{ - {\bf{1}}}}} \right) = det{\rm{ }}A\).
Compute the determinants of the elementary matrices given in Exercises 25-30. (See Section 2.2)
\[\left[ {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{1}}&{\bf{0}}\\{\bf{0}}&k&{\bf{1}}\end{aligned}} \right]\]
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