Question

If A = `[("a", "b"),("c", "d")]` and I = `[(1, 0),(0, 1)]` show that A² – (a + d)A = (bc – ad)I₂

Answer 1

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