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Chapter 3: Q20E (page 165)

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.

\(\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{a + kc}&{b + kd}\\c&d\end{array}} \right]\)

Short Answer

Expert verified

When the rows are swapped, the determinant changes.

Step by step solution

01

Find the determinant of the first matrix

The determinant of the matrix \(\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\) can be calculated as shown below:

\(\left| {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right| = ad - bc\)

02

Find the determinant of the second matrix

The determinant of the matrix \(\left[ {\begin{array}{*{20}{c}}{a + kc}&{b + kd}\\c&d\end{array}} \right]\) can be calculated as shown below:

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}{a + kc}&{b + kd}\\c&d\end{array}} \right| = d\left( {a + kc} \right) - c\left( {b + kd} \right)\\ = da + dkc - bc - ckd\\ = ad - bc\end{array}\)

So, when the rows are swapped, the determinant changes.

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Most popular questions from this chapter

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. \(det\left( {A + B} \right) = det{\rm{ }}A + det{\rm{ }}B\).

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

23. \(\left( {\begin{aligned}{*{20}{c}}2&0&0&6\\1&{ - 7}&{ - 5}&0\\3&8&6&0\\0&7&5&4\end{aligned}} \right)\)

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.

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

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

\(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{ - {\bf{7}}}&{\bf{3}}&{ - {\bf{5}}}\\{\bf{0}}&{\bf{0}}&{\bf{2}}&{\bf{0}}&{\bf{0}}\\{\bf{7}}&{\bf{3}}&{ - {\bf{6}}}&{\bf{4}}&{ - {\bf{8}}}\\{\bf{5}}&{\bf{0}}&{\bf{5}}&{\bf{2}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{0}}&{\bf{9}}&{ - {\bf{1}}}&{\bf{2}}\end{array}} \right|\)
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