12. (a) Let A =(2 12 1)∈ M2,2(Z). By nding a suitable matrix B, show thatA is a zero-divisor. [4 marks℄(b) Let a, b, c b e distin t elements of an integral domain su h that ca = bc.Show that c = 0.
Question
- (a) Let A =(2 12 1)∈ M2,2(Z). By nding a suitable matrix B, show thatA is a zero-divisor. [4 marks℄(b) Let a, b, c b e distin t elements of an integral domain su h that ca = bc.Show that c = 0.
Solution
(a) A matrix A is a zero-divisor if there exists a non-zero matrix B such that the product AB or BA is the zero matrix.
The given matrix A is (2 12 1). However, this is not a 2x2 matrix. A 2x2 matrix should have 4 elements. Please provide the correct matrix.
(b) An integral domain is a non-zero commutative ring in which the product of any two non-zero elements is always non-zero.
Given that ca = bc for distinct elements a, b, c in the integral domain, and a ≠ b, we can rearrange this equation to give c(a-b) = 0.
In an integral domain, if the product of two quantities is zero, then at least one of the quantities must be zero. Since a and b are distinct, a-b ≠ 0, so it must be that c = 0.
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