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Question
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find A matrix Z such that X + Y + Z is a zero matrix
Solution
Given that X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`
X + Y + Z = 0
⇒ `[(3, 1, -1),(5, -2, -3)] + [(2, 1, -1),(7, 2, 4)] + [("a", "b", "c"),("d", "e", "f")] = [(0, 0, 0),(0, 0, 0)]`
Where Z = `[("a", "b", "c"),("d", "e", "f")]`
⇒ `[(3 + 2 + "a", 1 + 1 + "b", -1 - 1 + "c"),(5 + 7 + "d", -2 + 2 + "e", -3 + 4 + "f")] = [(0, 0, 0),(0, 0, 0)]`
⇒ `[(5 + "a", 2 + "b", -2 + "c"),(12 + "d", "e", 1 + "f")] = [(0, 0, 0),(0, 0, 0)]`
Equating the corresponding elements, we get
5 + a = 0
⇒ a = – 5, 2 + b = 0
⇒ b = – 2 – 2 + c = 0
⇒ c = 2
12 + d = 0
⇒ d = – 12, e = 0, 1 + f = 0
⇒ f = – 1
Hence, the matrix Z = `[(-5, -2, 2),(-12, 0, -1)]`
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