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

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Answer 1 · 2017-05-21T22:14:41

See 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.