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Question
If A = `[(1, 5),(7, 12)]` and B `[(9, 1),(7, 8)]`, find a matrix C such that 3A + 5B + 2C is a null matrix.
Solution
Order of matrices A and B is 2 × 2.
∴ Order of matrix C must be 2 × 2.
Let C = `[("a", "b"),("c", "d")]`
∴ 3A + 5B + 2C = 0
⇒ `3[(1, 5),(7, 12)] + 5[(9, 1),(7, 8)] + 2[("a", "b"),("c", "d")] = [(0, 0),(0, 0)]`
⇒ `[(3, 15),(21, 36)] + [(45, 5),(35, 40)] + [(2"a", 2"b"),(2"c", 2"")] = [(0, 0),(0, 0)]`
⇒ `[(3 + 45 + 2"a", 15 + 5 + 2"b"),(21 + 35 + 2"c", 36 + 40 + 2"d")] = [(0, 0),(0, 0)]`
⇒`[(48 + 2"a", 20 + 2"b"),(56 + 2"c", 76 + 2"d")] = [(0, 0),(0, 0)]`
Equating the corresponding elements, we get,
48 + 2a = 0
⇒ 2a = – 48
⇒ a = – 24
20 + 2b = 0
⇒ 2b = – 20
⇒ b = – 10
56 + 2c = 0
⇒ 2c = – 56
⇒ c = – 28
76 + 2d = 0
⇒ 2d = – 76
⇒ d = – 38
Hence, C = `[(-2, -10),(-2, -38)]`
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