Question

If A = `[(3, 5)]`, B = `[(7, 3)]`, then find a non-zero matrix C such that AC = BC.

Answer 1

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.