Question

How can we represent a given matrix as the product of two other matrices?

Answer 1

See explanation...

Explanation:

Given a matrix `M`, choose any invertible matrix `A` of appropriate dimensions. Then:

`M = IM = (A A^(-1))M = A (A^(-1)M)`

For example, given `M=((1,2),(3,4))` we can choose `A=((1, 1),(0, 1))`.

Then `A^(-1) = ((1, -1),(0, 1))`

So:

`A^(-1)M = ((1, -1),(0, 1))((1, 2),(3, 4)) = ((-2, -2),(3, 4))`

and we find:

`((1, 1),(0, 1))((-2, -2),(3, 4)) = ((1, 2),(3, 4))`

as required.