Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 21E from Chapter 3.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 21E
Chapter:
Problem:
In Exercises 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.
Step-by-Step Solution
Given Information
We are given with following matrices. \[ \left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right] , \left[ \begin{array} { c c } { a } & { b } \\ { k c } & { k d } \end{array} \right] \] We have to explore the effect of an elementary row operation on determinant of matrix
Step-1:
Determinant of first matrix: \[\left| {\begin{array}{*{20}{l}} a&b\\ c&d \end{array}} \right| = ad - bc\] Determinant of second matrix: \[ \begin{aligned} \left| \begin{array} { l l } { a } & { b } \\ { k c } & { k d } \end{array} \right| & = a k d - b k c \\ & = k ( a d - b c ) \end{aligned} \] When the row 2 multiplied by k, then the determinant is multiplied by k.
The determinant is multiplied by k. when a scalar k is multiplied to one of the rows.
We are given with following matrices. \[ \left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right] , \left[ \begin{array} { c c } { a } & { b } \\ { k c } & { k d } \end{array} \right] \] We have to explore the effect of an elementary row operation on determinant of matrix
Step-1:
Determinant of first matrix: \[\left| {\begin{array}{*{20}{l}} a&b\\ c&d \end{array}} \right| = ad - bc\] Determinant of second matrix: \[ \begin{aligned} \left| \begin{array} { l l } { a } & { b } \\ { k c } & { k d } \end{array} \right| & = a k d - b k c \\ & = k ( a d - b c ) \end{aligned} \] When the row 2 multiplied by k, then the determinant is multiplied by k.
The determinant is multiplied by k. when a scalar k is multiplied to one of the rows.