If the matrix AB = 0, then a = 0 or B = 0? Is it correct If not, why not? It's better to have counter examples

If the matrix AB = 0, then a = 0 or B = 0? Is it correct If not, why not? It's better to have counter examples

Incorrect. Counterexample:
A =
0 0
0 1
B =
1 0
0 0
Then AB = Ba = 0, but a and B are not 0

Why do matrices A and B satisfy AB = 0 and | a | ≠ 0 when B = 0?

DET (a) ≠ 0 means that a is nonsingular and therefore reversible
B = 0 can be obtained by multiplying a ^ (- 1) left by ab = 0

Three node matrix A = | 0 01 |, and a is similar to B, R (AB-A) =? | 0 10 | 1 00|

Although this question is small, it has rich knowledge
A is reversible, and its eigenvalues are 1,1, - 1, and diagonalizable
Because a and B are similar, the eigenvalues of B are 1,1, - 1 and can be diagonalized
So the eigenvalues of B-E are 0,0, - 2
So r (B-E) = 1
So r (AB-A) = R [a (B-E)] = R (B-E) = 1