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Question
If A = `[(3, 5)]`, B = `[(7, 3)]`, then find a non-zero matrix C such that AC = BC.
Solution
We have, A = `[(3, 5)]_(1 xx 2)` and B = `[(7, 3)]_(1 xx 2)`
For AC = BC
We have order of C = 2 × n
For n = 1
Let C = `[(x),(y)]`
∴ AC = `[(3, 5)] [(x),(y)] = [(3x + 5y]`
And BC = `[(7, 3)] [(x),(y)]` = [3x + 5y]
For AC = BC,
[3x + 5y] = [7x + 3y]
⇒ 3x + 5y = 7x + 3y
⇒ 4x = 2y
⇒ x = `1/2 y`
⇒ y = 2x
∴ C = `[(x),(2x)]`
We see that on taking C of order 2 × 1, 2 × 2, 2 × 3, ..., we get
C = `[(x),(2x)], [(x, x),(2x, 2x)], [(x, x, x),(2x, 2x, 2x)]`...
In general,
C = `[("k"),(2"k")], [("k", "k"),(2"k", 2"k")]` etc ...
Where, k is any real number.
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