Find nonzero matrices ABC such that A is not equal to B and

Find nonzero matrices A,B,C such that A is not equal to B and AC =BC

Solution

For, AC = BC

C must be singular(noninvertible) matrix i.e the determinant must be zero.

If we multiply {{a,b},{c,d}} to C then the result will be {{a+b,a+b},{c+d,c+d}}.

So, we have to choose two matrices A and B such that their row-sums are individually equal.

So, A = {{1,0},{0,1}},

B = {{0,1},{1,0}}, and

C = {{1,1},{1,1}}