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Question
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M
Solution
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}`
Closure axiom:
Take A = `((x, x),(x, x)) ∈ "M"`
B = `((y, y),(y, y)) ∈ "M"`
Then AB = `((2xy, 2xy),(2xy, 2xy))`
Since x ≠ 0, y ≠ 0
We see that 2xy ≠ 0 and So AB ∈ M.
This shows that M is closed under matrix multiplication.
Commutative axiom:
AB = `((2xy, 2xy),(2xy, 2xy))`
= BA for all A, B ∈ M
Here, Matrix multiplication is commutative ......(though in general, matrix multiplication is not commutative)
Associative axiom:
Since matrix multiplication is associative, this axiom holds goods for M.
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