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Question
If A = `[(9 , 1),(7 , 8)]` , B = `[(1 , 5),(7 , 12)]`
find matrix C such that 5A + 5B + 2C is a null matrix.
Solution
Let C = `[(a , b),(c ,d)]`
We have A = `[(9 , 1),(7 , 8)]` , B = `[(1 , 5),(7 , 12)]`
Now 5A + 3B + 2C = 0
⇒ `5[(9 , 1),(7 , 8)]+3 [(1 , 5),(7 , 12)]+2[(a , b),(c , d)] = [(0 , 0),(0 , 0)]`
⇒ `[(45 , 5),(35 , 40)]+ [(3 , 15),(21 , 36)]+[(2a , 2b),(2c , 2d)] = [(0 , 0),(0 , 0)]`
⇒ `[(45 + 3 + 2a, 5 + 15 + 2b),(35 + 21 + 2c, 40 + 36 + 2d)] = [(0 , 0),(0 , 0)]`
⇒ `[(48 + 2a , 20 + 2b),(56 + 2c , 76 + 2d)] = [(0 , 0),(0 , 0)]`
⇒ 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
Thus C = `[(-24 , -10),(-28, -38)]`.
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