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.
Suppose that we are given an m×n matrix A and a real number k and that
Elementwise this looks like
for each element aij of matrix A.
Here, we are multiplying two real numbers and the result is zero, so one of the must be zero.
If k=0 we are done.
If k≠0 this means that aij=0 for each element of matrix A, so A=0.