   Chapter 1.6, Problem 8E

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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() or not and complete the multiplication table for S, where S={I,A,B,C}

I=,A=,B=,C=

Explanation

Given information:

S={I,A,B,C}

I=,A=,B=,C=

Given table is:

IABCIABBCIAC

Formula used:

Definition: Matrix multiplication

The product of m×n matrix A over and n×p matrix B over 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:

columnjofBcolumnjofCrowiofA[ai1ai2ai3ain][b1jb2jb3jbnj]=[cij]rowiofC

Where cij=ai1b1j+ai2b2j++ainbnj.

Explanation:

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

I=,A=,B=,C=

The given multiplication table of S is

IABCIABBCIAC

It is given that BI=B,BA=C,BB=I,BC=A

First let us complete the table,

(i) II=

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]

=

=I

(ii) IA=

By using definition of matrix multiplication,

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

=[0+01+00+10+0]

=

=A

(iii) IB=

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+001]

=

=B

(iv) IC=

By using definition of matrix multiplication,

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

=[0+01+0010+0]

=

=C

(v) AI=[011&

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