Page 50 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes which is zero element in R.
Thus R is a ring with zero divisors.
It can also be verified that R is also a commutative ring with unity element (1, 1).

Example 22: Prove that M the set of all 2 × 2 matrices of the form

a bi c di
, i 1
c di a bi
where a, b, c, d are real numbers, form a division ring.

1 0
Solution: Since I = M is a ring with unity under matrix addition and multiplication.
0 1
Let A be a non-zero matrix in M, and let

a bi c di
A
c di a bi

where a, b, c, d are not all zero. Consider

a bi c di
B a b 2 c 2 d 2 a 2 2 c 2 d 2
c di a bi
a 2 b 2 c 2 d 2 a 2 b 2 c 2 d 2

1 0
Evidently B M . Also A B = B A = .
0 1
Thus every non-zero matrix of M is invertible. Hence M is a division ring.

Example 23: Prove that the set of integers is a subring of the ring of rational numbers.
Solution: Let I be the set of integers and Q the set of rational numbers.

Clearly I Q and ,a b I a b I and a b I

Therefore, I is a subring of Q.

a b
Example 24: Show that the set of matrices is a subring of the ring of 2 × 2 matrices
0 c
with integral elements.
a b
Solution: Let M be the set of matrices of the type
0 c
Clearly M R

a b a b
Let A 1 1 , B 2 2 M then
0 c 0 c
1 2
a a b b
A B 1 2 1 2 M and
0 c c
1 2

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