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

We have, A=`[3 5]_(1xx2)` and B=`[7 3]_(1xx1)`
Let `C=[{:(x),(y):}]_(2xx1)` is a non-zero matrix of order `2xx1`.
`therefore AC=[3 5][{:(x),(y):}]=[3x+5y]`
and `BC=[7 3][{:(x),(y):}]=[7x+3y]`
For AC=BC,
`[3x+5y ]=[7x+3y]`
On using equality of matrix we get
` 3x+5y=7x+3y`
`rArr 4x=2y`
`rArr x=(1)/(2)y`
`rArr y=2x`
`rArr y=2x`
`therefore C=[{:(x),(2x):}]`
We see that on taking C of order `2xx1.2xx2.2xx3`....... we get
`C=[{:(x),(2x):}][{:(x,x),(2x,2x):}][{:(x,x,x),(2x,2x,2x):}]`
In general,
`C=[{:(k) ,(2k):}][{:(k,k),(2k,2k):}]` etc
where, k is any real number