Elements Of Modern Algebra

8th Edition

Gilbert + 2 others

ISBN: 9781285463230

Textbook Problem

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Let S be the set of four matrices S = { I , A , B , C } , where I = [ 1 0 0 1 ] , A = [ 0 − 1 1 0 ] , B = [ − 1 0 0 − 1 ] , C = [ 0 1 − 1 0 ] .

Follow the procedure described in Exercise 3 of section 1.4 to complete the following multiplication table for S . (In this case, the product B C = A is entered as shown in the row with B at the left end and in the column with C at the top.) Is S closed under multiplication in M 2 ( ℝ ) ?

⋅ I A B C I A B B C I A C

To determine

Whether S is closed under multiplication in M2(ℝ) or not and complete the multiplication table for S, where S={I, A, B, C}

I=[1001], A=[0−11 0], B=[−1 0 0−1], C=[ 01−10]

Explanation

Given information:

S={I, A, B, C}

I=[1001], A=[0−11 0], B=[−1 0 0−1], C=[ 01−10]

Given table is:

⋅IABCIABBCIAC

Formula used:

Definition: Matrix multiplication

The product of m×n matrix A over ℝ and n×p matrix B over ℝ is m×p matrix C=AB, where the element cij in row i and column j of AB is found by using the elements in row i of A, and the elements in column j of B in the following manner:

column j of B column j of Crow i of A[⋮⋮⋮⋮ai1ai2ai3⋯ain⋮⋮⋮⋮]⋅[⋯b1j⋯⋯b2j⋯⋯b3j⋯⋮⋯bnj⋯]=[⋮⋯cij⋯⋮]row i of C

Where cij=ai1b1j+ai2b2j+ ⋯ + ainbnj.

Explanation:

Let S be the set of four matrices S={I, A, B, C}

I=[1001], A=[0−11 0], B=[−1 0 0−1], C=[ 01−10]

The given multiplication table of S is

⋅IABCIABBCIAC

It is given that B⋅I=B, B⋅A=C, B⋅B=I, B⋅C=A

First let us complete the table,

(i) I⋅I=[1001][1001]

By using definition of matrix multiplication,

=[(1)(1)+(0)(0)(1)(0)+(0)(1)(0)(1)+(1)(0)(0)(0)+(1)(1)]

=[1+00+00+00+1]

=[1001]

(ii) I⋅A=[1001][0−11 0]

By using definition of matrix multiplication,

=[(1)(0)+(0)(1)(1)(−1)+(0)(0)(0)(0)+(1)(1)(0)(−1)+(1)(0)]

=[0+0−1+00+1 0+0]

=[0−11 0]

(iii) I⋅B=[1001][−1 0 0−1]

By using definition of matrix multiplication,

=[(1)(−1)+(0)(0)(1)(0)+(0)(−1)(0)(−1)+(1)(0)(0)(0)+(1)(−1)]

=[−1+0 0+0 0+00−1]

=[−1 0 0−1]

(iv) I⋅C=[1001][ 01−10]

By using definition of matrix multiplication,

=[(1)(0)+(0)(−1)(1)(1)+(0)(0)(0)(0)+(1)(−1)(0)(1)+(1)(0)]

=[0+0 1+00−10+0]

=[ 01−10]

(v) A⋅I=[0−11&

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Sect-1.1 P-1TFE Sect-1.1 P-2TFE Sect-1.1 P-3TFE Sect-1.1 P-4TFE Sect-1.1 P-5TFE Sect-1.1 P-6TFE Sect-1.1 P-7TFE Sect-1.1 P-8TFE Sect-1.1 P-9TFE Sect-1.1 P-10TFE

Sect-1.1 P-1E Sect-1.1 P-2E Sect-1.1 P-3E Sect-1.1 P-4E Sect-1.1 P-5E Sect-1.1 P-6E Sect-1.1 P-7E Sect-1.1 P-8E Sect-1.1 P-9E Sect-1.1 P-10E Sect-1.1 P-11E Sect-1.1 P-12E Sect-1.1 P-13E Sect-1.1 P-14E Sect-1.1 P-15E Sect-1.1 P-16E Sect-1.1 P-17E Sect-1.1 P-18E Sect-1.1 P-19E Sect-1.1 P-20E Sect-1.1 P-21E Sect-1.1 P-22E Sect-1.1 P-23E Sect-1.1 P-24E Sect-1.1 P-25E Sect-1.1 P-26E Sect-1.1 P-27E Sect-1.1 P-28E Sect-1.1 P-29E Sect-1.1 P-30E Sect-1.1 P-31E Sect-1.1 P-32E Sect-1.1 P-33E Sect-1.1 P-34E Sect-1.1 P-35E Sect-1.1 P-36E Sect-1.1 P-37E Sect-1.1 P-38E Sect-1.1 P-39E Sect-1.1 P-40E Sect-1.1 P-41E Sect-1.1 P-42E Sect-1.1 P-43E Sect-1.2 P-1TFE Sect-1.2 P-2TFE Sect-1.2 P-3TFE Sect-1.2 P-4TFE Sect-1.2 P-5TFE Sect-1.2 P-6TFE Sect-1.2 P-7TFE Sect-1.2 P-8TFE Sect-1.2 P-9TFE Sect-1.2 P-1E Sect-1.2 P-2E Sect-1.2 P-3E Sect-1.2 P-4E Sect-1.2 P-5E Sect-1.2 P-6E Sect-1.2 P-7E Sect-1.2 P-8E Sect-1.2 P-9E Sect-1.2 P-10E Sect-1.2 P-11E Sect-1.2 P-12E Sect-1.2 P-13E Sect-1.2 P-14E Sect-1.2 P-15E Sect-1.2 P-16E Sect-1.2 P-17E Sect-1.2 P-18E Sect-1.2 P-19E Sect-1.2 P-20E Sect-1.2 P-21E Sect-1.2 P-22E Sect-1.2 P-23E Sect-1.2 P-24E Sect-1.2 P-25E Sect-1.2 P-26E Sect-1.2 P-27E Sect-1.2 P-28E Sect-1.3 P-1TFE Sect-1.3 P-2TFE Sect-1.3 P-3TFE Sect-1.3 P-4TFE Sect-1.3 P-5TFE Sect-1.3 P-6TFE Sect-1.3 P-1E Sect-1.3 P-2E Sect-1.3 P-3E Sect-1.3 P-4E Sect-1.3 P-5E Sect-1.3 P-6E Sect-1.3 P-7E Sect-1.3 P-8E Sect-1.3 P-9E Sect-1.3 P-10E Sect-1.3 P-11E Sect-1.3 P-12E Sect-1.4 P-1TFE Sect-1.4 P-2TFE Sect-1.4 P-3TFE Sect-1.4 P-4TFE Sect-1.4 P-5TFE Sect-1.4 P-6TFE Sect-1.4 P-7TFE Sect-1.4 P-8TFE Sect-1.4 P-9TFE Sect-1.4 P-1E Sect-1.4 P-2E Sect-1.4 P-3E Sect-1.4 P-4E Sect-1.4 P-5E Sect-1.4 P-6E Sect-1.4 P-7E Sect-1.4 P-8E Sect-1.4 P-9E Sect-1.4 P-10E Sect-1.4 P-11E Sect-1.4 P-12E Sect-1.4 P-13E Sect-1.4 P-14E Sect-1.4 P-15E Sect-1.4 P-16E Sect-1.5 P-1TFE Sect-1.5 P-2TFE Sect-1.5 P-3TFE Sect-1.5 P-1E Sect-1.5 P-2E Sect-1.5 P-3E Sect-1.5 P-4E Sect-1.5 P-5E Sect-1.5 P-6E Sect-1.5 P-7E Sect-1.5 P-8E Sect-1.5 P-9E Sect-1.5 P-10E Sect-1.6 P-1TFE Sect-1.6 P-2TFE Sect-1.6 P-3TFE Sect-1.6 P-4TFE Sect-1.6 P-5TFE Sect-1.6 P-6TFE Sect-1.6 P-7TFE Sect-1.6 P-8TFE Sect-1.6 P-9TFE Sect-1.6 P-10TFE Sect-1.6 P-11TFE Sect-1.6 P-12TFE Sect-1.6 P-1E Sect-1.6 P-2E Sect-1.6 P-3E Sect-1.6 P-4E Sect-1.6 P-5E Sect-1.6 P-6E Sect-1.6 P-7E Sect-1.6 P-8E Sect-1.6 P-9E Sect-1.6 P-10E Sect-1.6 P-11E Sect-1.6 P-12E Sect-1.6 P-13E Sect-1.6 P-14E Sect-1.6 P-15E Sect-1.6 P-16E Sect-1.6 P-17E Sect-1.6 P-18E Sect-1.6 P-19E Sect-1.6 P-20E Sect-1.6 P-21E Sect-1.6 P-22E Sect-1.6 P-23E Sect-1.6 P-24E Sect-1.6 P-25E Sect-1.6 P-26E Sect-1.6 P-27E Sect-1.6 P-28E Sect-1.6 P-29E Sect-1.6 P-30E Sect-1.6 P-31E Sect-1.6 P-32E Sect-1.7 P-1TFE Sect-1.7 P-2TFE Sect-1.7 P-3TFE Sect-1.7 P-4TFE Sect-1.7 P-5TFE Sect-1.7 P-6TFE Sect-1.7 P-1E Sect-1.7 P-2E Sect-1.7 P-3E Sect-1.7 P-4E Sect-1.7 P-5E Sect-1.7 P-6E Sect-1.7 P-7E Sect-1.7 P-8E Sect-1.7 P-9E Sect-1.7 P-10E Sect-1.7 P-11E Sect-1.7 P-12E Sect-1.7 P-13E Sect-1.7 P-14E Sect-1.7 P-15E Sect-1.7 P-16E Sect-1.7 P-17E Sect-1.7 P-18E Sect-1.7 P-19E Sect-1.7 P-20E Sect-1.7 P-21E Sect-1.7 P-22E Sect-1.7 P-23E Sect-1.7 P-24E Sect-1.7 P-25E Sect-1.7 P-26E Sect-1.7 P-27E Sect-1.7 P-28E Sect-1.7 P-29E Sect-1.7 P-30E

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Elements Of Modern Algebra

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Elements Of Modern Algebra

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Algebra Elements Of Modern Algebra 8th Edition