Could somebody answer part 3 of this question Thank you C Ri

Could somebody answer part (3) of this question?

Thank you

C. Ring of 2 X 2 Matrices Let M 2(R) designate the set of all 2 x 2 matrices whose entries are real numbers a. b, c, and d, with the following addition and multiplication: 1 Verify that M 2(R) satisfies the ring axioms. 2 Show that .M2(R) is not commutative and has a unity. 3 Explain why M2(R) is not an integral domain or a field.

Solution

The additive identity for this set is unique as

0   0

0   0

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Multiplicative identity is I2

1 0

0   1 is unique.

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For any element A in M2, A*0 = 0*A = 0 matrix

For any A

a b

c d    there exists -A = -a -b

                                   -c -d

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If 0 =1 i.e. if multiplicative inverse = identity matrix means then there is only one element in the ring that is 0 matrix

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Let us check the ring axioms

Matrix addition is associative

Matrix addition is commutative

Additive identity (0) matrix and _A additive inverse exists.

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But M2 is not commutative under multiplication as matrix multiplication is not commutative.

BA not equals AB.

BUt for non singular matrices inverse exists and I2 exists.

3) M2 is not an integral domain or field because.

xy =0 should prove x =0 y =0

BUt a matrix AB may be singular though A and B are not singular.

Hence not integral domain or field