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Copyright 2021 Pearson Education, Inc. 2. Groups 18. Matrix multiplication is associative, so it remains to show that G is closed under matrix multiplication, G has an identity and each element of G has an inverse. The e a b table for G is e b from which all of these properties are easily spotted. e a a a e b b e a 19. a. We must show that S is closed under *, that is, that a+b+ab -1 for a, beS. Now a + b+ ab = -1 if and only if 0 = ab+ a+b+1= (a+1)(b+1). This is the case if and only if either a =-1 or b =-1, which is not the case for a, be S. b. Associative: We have a * (b *c) = a * (b+c+bc) = a + (b+c+ bc)+ a(b+c+ bc) = a+b+c+ ab+ ac + be + abc MacBook Pro G Search or type URL

Copyright 2021 Pearson Education, Inc. 2. Groups 18. Matrix multiplication is associative, so it remains to show that G is closed under matrix multiplication, G has an identity and each element of G has an inverse. The e a b table for G is e b from which all of these properties are easily spotted. e a a a e b b e a 19. a. We must show that S is closed under *, that is, that a+b+ab -1 for a, beS. Now a + b+ ab = -1 if and only if 0 = ab+ a+b+1= (a+1)(b+1). This is the case if and only if either a =-1 or b =-1, which is not the case for a, be S. b. Associative: We have a * (b *c) = a * (b+c+bc) = a + (b+c+ bc)+ a(b+c+ bc) = a+b+c+ ab+ ac + be + abc MacBook Pro G Search or type URL

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.5: The Binomial Theorem
Problem 13E
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Q18 picture 2 is the general answer key. Do not use the table on the answer key. write speific steps for this question to prove it is under operation and it is a group
pdf Q 山☆ K Paused 17 / 22| 129% Inverse: The product property 1-UC(1,)–de( 7A A)–Uet(A )*Uet(A shows that if det(4) = 1, then det(A-) = 1 also. Copyright O 2021 Pearson Education, Inc. 2. Groups 9. 18. Matrix multiplication is associative, so it remains to show that G is closed under matrix multiplication, G has an identity and each element of G has an inverse. The * e a table for G is e from which all of these properties are easily spotted. e a a e b b e a We must show that S is closed under *, that is, that a+b+ab±-1 for a, be S. Now a + b + ab = –1 if and only if 0 = ab+ a+b+1= (a+1)(b+1). This is the case if and only if either a = -1 or b = -1, which is not the case for a, bɛ S. 19. а. b. Associative: We have a * (b * c)= a * (b+c+bc)% = a + (b+c+ bc)+a(b+c + bc) = a +b+c + ab + ac + bc + abc MacBook Pro Search or type URL & *..
Transcribed Image Text:pdf Q 山☆ K Paused 17 / 22| 129% Inverse: The product property 1-UC(1,)–de( 7A A)–Uet(A )*Uet(A shows that if det(4) = 1, then det(A-) = 1 also. Copyright O 2021 Pearson Education, Inc. 2. Groups 9. 18. Matrix multiplication is associative, so it remains to show that G is closed under matrix multiplication, G has an identity and each element of G has an inverse. The * e a table for G is e from which all of these properties are easily spotted. e a a e b b e a We must show that S is closed under *, that is, that a+b+ab±-1 for a, be S. Now a + b + ab = –1 if and only if 0 = ab+ a+b+1= (a+1)(b+1). This is the case if and only if either a = -1 or b = -1, which is not the case for a, bɛ S. 19. а. b. Associative: We have a * (b * c)= a * (b+c+bc)% = a + (b+c+ bc)+a(b+c + bc) = a +b+c + ab + ac + bc + abc MacBook Pro Search or type URL & *..
S Ga Un G de 土 Do So 00000000000.. O ☆ 司 olatform.virdocs.com/r/s/0/doc/1559577/sp/165492540/mi/542334716?cfi=%2F4%2F2% Sa group b. Show that (nZ, +) ~ (Z, +). 28 In Exercises 11 0 through 180, determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries lie on the main diagonal, from the upper left to the lower right corner. An upper-triangular matrix is a square matrix with only zero entries below the main diagonal. Associated with each n x n matrix A is a number called the determinant of A, denoted by det(A). If A and B are both n x n matrices, then det (AB) =det (A) det (B). Also, det(In) = 1 and A is invertible if and only if det (A) +0. 11. All n x n diagonal matrices under matrix addition. 12. All n x n diagonal matrices under matrix multiplication. 13. All n x n diagonal matrices with no zero diagonal entry under matrix multiplication. 14. All n x n diagonal matrices with all diagonal entries 1 or –1 under matrix multiplication. 15. All n x n upper-triangular matrices under matrix multiplication. 16. All n x n upper-triangular matrices under matrix addition 17. All n x n upper-triangular matrices with determinant 1 under matrix multiplication. 1 V3 2 , and 1 18. The set of 2 × 2 matrices G = {e, a, b} where e = a = V3 2 1 V3 b = 2 under matrix multiplication. V3 1 2 19. Let S be the set of all real numbers except -1. Define * on S by 人 MacBook Pro G Search or type URL %23 & 3 6 8. 9. E T Y S F G J K L + R w/
Transcribed Image Text:S Ga Un G de 土 Do So 00000000000.. O ☆ 司 olatform.virdocs.com/r/s/0/doc/1559577/sp/165492540/mi/542334716?cfi=%2F4%2F2% Sa group b. Show that (nZ, +) ~ (Z, +). 28 In Exercises 11 0 through 180, determine whether the given set of matrices under the specified operation, matrix addition or multiplication, is a group. Recall that a diagonal matrix is a square matrix whose only nonzero entries lie on the main diagonal, from the upper left to the lower right corner. An upper-triangular matrix is a square matrix with only zero entries below the main diagonal. Associated with each n x n matrix A is a number called the determinant of A, denoted by det(A). If A and B are both n x n matrices, then det (AB) =det (A) det (B). Also, det(In) = 1 and A is invertible if and only if det (A) +0. 11. All n x n diagonal matrices under matrix addition. 12. All n x n diagonal matrices under matrix multiplication. 13. All n x n diagonal matrices with no zero diagonal entry under matrix multiplication. 14. All n x n diagonal matrices with all diagonal entries 1 or –1 under matrix multiplication. 15. All n x n upper-triangular matrices under matrix multiplication. 16. All n x n upper-triangular matrices under matrix addition 17. All n x n upper-triangular matrices with determinant 1 under matrix multiplication. 1 V3 2 , and 1 18. The set of 2 × 2 matrices G = {e, a, b} where e = a = V3 2 1 V3 b = 2 under matrix multiplication. V3 1 2 19. Let S be the set of all real numbers except -1. Define * on S by 人 MacBook Pro G Search or type URL %23 & 3 6 8. 9. E T Y S F G J K L + R w/
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