Prove the property. Let 1 + a 1 1 1 + a 1 1 A = 1 1 + b 1 1 1 + b 1 1 + a 1 1 1 1 1 + c 1 1 + b 1 1 1 1 + C 0 0 -a 1 Since the determinant of a matrix obtained from adding multiples of a row to another row in the matrix is equal to the determinant of the original matrix, we can reduce the amount of work needed to expand this determinant by using the elementary row operation of row addition. Using the third row to reduce the first and second entries in the first column to zero gives the following. Complete the proof by computing and factoring the determinant of the reduced matrix. b 1 1 1 + c 1 = abc 1 + = ac bl 1 + c ,a # 0, b# 0, c = 0. = ac + ab + bc + abc ac + ab + bc + + abc b = abc ( 1 + 1 + 1 + ¹ ) a + a 0, b0, c# 0
Prove the property. Let 1 + a 1 1 1 + a 1 1 A = 1 1 + b 1 1 1 + b 1 1 + a 1 1 1 1 1 + c 1 1 + b 1 1 1 1 + C 0 0 -a 1 Since the determinant of a matrix obtained from adding multiples of a row to another row in the matrix is equal to the determinant of the original matrix, we can reduce the amount of work needed to expand this determinant by using the elementary row operation of row addition. Using the third row to reduce the first and second entries in the first column to zero gives the following. Complete the proof by computing and factoring the determinant of the reduced matrix. b 1 1 1 + c 1 = abc 1 + = ac bl 1 + c ,a # 0, b# 0, c = 0. = ac + ab + bc + abc ac + ab + bc + + abc b = abc ( 1 + 1 + 1 + ¹ ) a + a 0, b0, c# 0
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.1: The Determinants Of A Matrix
Problem 70E: The determinant of a 22 matrix involves two products. The determinant of a 33 matrix involves six...
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Transcribed Image Text:Prove the property.
Let
1 + a
1
1
1 + a
1
1
A =
1
1 + b
1
1
1 + b
1
1 + a
1
1
1
1
1 + c
1
1 + b
1
1
1
1 + C
0
0 -a
1
Since the determinant of a matrix obtained from adding multiples of a row to another row in the matrix is equal to the determinant of the original matrix, we can reduce the amount of work needed to expand this
determinant by using the elementary row operation of row addition. Using the third row to reduce the first and second entries in the first column to zero gives the following. Complete the proof by computing and
factoring the determinant of the reduced matrix.
b
1
1
1 + c
1
= abc 1 +
= ac bl
1 + c
,a # 0, b# 0, c = 0.
= ac + ab + bc +
abc ac + ab + bc +
+
abc
b
= abc ( 1 + 1 + 1 + ¹ )
a
+
a 0, b0, c# 0
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