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Question
If A = `[("a", "b"),("c", "d")]` and I = `[(1, 0),(0, 1)]` show that A2 – (a + d)A = (bc – ad)I2
Solution
A = `[("a", "b"),("c", "d")]`, I = `[(1, 0),(0, 1)]`
A2 = `[("a", "b"),("c", "d")] xx [("a", "b"),("c", "d")]`
= `[("a"^2 + "bc", "ab" + "bd"),("ac" + "dc", "bc" + "d"^2)]`
L.H.S. = A2 – (a + d)A
= `[("a"^2 + "bc", "ab" + "bd"),("ac" + "cd", "bc" + "d"^2)] - ("a" + "d")[("a", "b"),("c","d")]`
= `[("a"^2 + "bc", "ab" + "bd"),("ac" + "cd", "bc" + "d"^2)] - [("a"^2 + "ad", "ab" + "bd"),("ac" + "cd", "ad" + "d"^2)]`
= `[("bc" - "ad", 0),(0, "bc" - "ad")]`
= `("bc" - "ad") [(1, 0),(0, 1)]`
= (bc – ad)I
L.H.S. = R.H.S.
A2 – (a + d)A = (bc – ad)I2
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