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Chapter 3: Q35Q (page 165)

In Exercise 33-36, verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)where E is the elementary matrix shown and \(A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\).

35. \(\left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\)

Short Answer

Expert verified

It is verified that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\).

Step by step solution

01

Determine matrix \(EA\)

It is given that \(A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],{\rm{ }}E = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\).

Compute matrix \(EA\) as shown below:

\[\begin{array}{c}EA = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{0 + c}&{0 + d}\\{a + 0}&{b + 0}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}c&d\\a&b\end{array}} \right]\end{array}\]

02

Verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)

The determinants of matrices Eand A are shown below:

\[\begin{array}{c}\det E = \left| {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right|\\ = 0 - 1\\ = - 1\\\det A = \left| {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right|\\ = ad - bc\end{array}\]

The determinant of matrix \(EA\) is shown below:

\(\begin{array}{c}\det EA = \left| {\begin{array}{*{20}{c}}c&d\\a&b\end{array}} \right|\\ = cb - da\\ = - 1\left( {ad - bc} \right)\\ = \left( {\det E} \right)\left( {\det A} \right)\end{array}\)

Thus, it is verified that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\).

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Most popular questions from this chapter

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.

In Exercises 21鈥�23, use determinants to find out if the matrix is invertible.

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

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.

\[\left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{1}}\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}k&{\bf{0}}&k\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right]\]

Find the determinants in Exercises 5-10 by row reduction to echelon form.

\(\left| {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{2}}&{ - {\bf{4}}}\\{\bf{0}}&{\bf{1}}&{\bf{2}}&{ - {\bf{5}}}\\{\bf{2}}&{\bf{7}}&{\bf{6}}&{ - {\bf{3}}}\\{ - {\bf{3}}}&{ - {\bf{10}}}&{ - {\bf{7}}}&{\bf{2}}\end{aligned}} \right|\)

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.

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