The solution of equations x+y+z=6,x+2y+3z=10,x+2y+λz=μ  has no solutions for

now,x+y+z=6x+2y+3z=10x+2y+λz=μ[A:B]=111⋮6 123⋮10 12λ⋮μ On applying R2→R2−R1 and R3→R3−R1=111:6012:401λ−1:μ−6On applying R3→R3−R2=111:6012:400λ−3:μ−10Rank (A) =2, when λ=3and Rank (AB) = 3, when μ≠10So, we can say that Rank (A) ≠ Rank (AB), when λ=3 and  μ≠10