   Chapter 1.6, Problem 22E

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# Let G be the set of all elements of M 2 ( ℝ ) that have one row that consists of zeros and one row of the form [ a       a ] , with a ≠ 0 .Show that G is closed under multiplication in M 2 ( ℝ ) .Show that for each x in G , there is an element y in G such that x y = y x = x .Show that G does not have an identity element with respect to multiplication.

(a)

To determine

To prove: Let G be the set of all elements of M2() that have one row that consists of zeros and one row of the form [aa], with a0, then G is closed under multiplication in M2().

Explanation

Given information:

G be the set of all elements of M2() that have one row that consists of zeros and one row of the form [aa], with a0.

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.

Proof:

Let G be the set of all elements of M2() that have one row that consists of zeros and one row of the form [aa], with a0.

Therefore, the elements of G are of the form, either A=[aa00] or B=[00bb], where a,b0 and a,b

(b)

To determine

To prove: Let G be the set of all elements of M2() that have one row that consists of zeros and one row of the form [aa], with a0, then for each x in G, there is an element y in G, such that xy=yx=x.

(c)

To determine

To prove: Let G be the set of all elements of M2() that have one row that consists of zeros and one row of the form [aa], with a0, then G does not have an identity element with respect to multiplication.

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