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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 31-36, mention an appropriate theorem in your explanation.
36. Find a formula for \(\det \left( {rA} \right)\) when \(A\) is an\(n \times n\) matrix.
Let \(u = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right]\), and \(v = \left[ {\begin{aligned}{*{20}{c}}c\\{\bf{0}}\end{aligned}} \right]\), where a, b, and c are positive integers (for simplicity). Compute the area of the parallelogram determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right]\), and \[\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right]\]. Draw a picture and explain what you find.
Find the determinant in Exercise 15, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].
15. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{{\bf{3g}}}&{{\bf{3h}}}&{{\bf{3i}}}\end{array}} \right|\]
In Exercise 19-24, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.
\(\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],\left[ {\begin{array}{*{20}{c}}a&b\\{kc}&{kd}\end{array}} \right]\)
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.
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