Let A be any mxn matrix and let 0 be the mxn matrix each...( linear algebra)

Question

Let A be any mxn matrix and let 0 be the mxn matrix each of whose entries is zero . Show that if kA=0 , then k=0 or A=0. 

 

I am very lost ! Please help me show this because I think its gonna be on my test. 

Answer 1

Let B = kA = 0 (the matrix 0)

 

Examine B_ij.

 

If B = 0, then B_ij = 0 for all i and j.

 

But B_ij = kA_ij

 

Then kA_ij = 0.

 

So either k = 0, or A_ij = 0 for all i and j.  But if A is any mxn matrix, then it is not true that A_ij = 0 for all i and j.  Therefore, k = 0.