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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 existence of identity, existence of inverse properties for the operation * on M
Solution
Identity axiom:
Let E = `(("e", "e"),("e", "e")) ∈ "M"` be such that
AR = A for any A ∈ M
Then AE = A gives `((x, x),(x, x)) (("e", "e"),("e", e")) = ((x, x),(x, x))`
.i.e. `((2x"e", 2x"e"),(2x"e", 2x"e")) = ((x, x),(x, x))` and so 2xe = x
⇒ e = `1/2`
Thus E = `((1/2, 1/2),(1/2, 1/2))` in M
Also, we can show that EA = A
Hence E is the identity element in M
Let B = `(("y", "y"),("y", "y")) ∈ "M"` be such that `((y, y),(y, y))` A = E
Then `((y, y),(y, y))((x, x),(x, x)) = ((1/2, 1/2),(1/2, 1/2))`
⇒ `((2xy, 2xy),(2xy, 2xy)) = ((1/2, 1/2),(1/2, 1/2))`
Hence 2xy = `1/2`
⇒ y = `1/(4x)`
This shows that BA = E
Similarly AB = E
i.e. B = `((1/(4x), 1/(4x)),(1/(4x), 1/(4x)))` is the inverse of A ∈ M
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