0

Using the definition of an inverse, can someone explain why $0_n$$_x$$_n$ cannot have an inverse.

Also can someone explain if AB=$0_n$$_x$$_n$ for two nxn nonzero matrices A and B, then how A nor B can have an inverse.

user6259845
  • 913
  • 1
    By definition, you cannot find an $n\times n$ matrix $A$ such that $0_{n\times n}A = A0_{n\times n}= I_{n\times n}$. – Chee Han Nov 17 '16 at 03:52
  • For the second one, suppose $A$ has an inverse $A^{-1}$, then left-multiplying $AB=0_{n\times n}$ with $A^{-1}$ gives $A^{-1}AB = B = A^{-1}0_{n\times n} = 0_{n\times n}$. This contradicts the fact that $B$ is a nonzero matrix. – Chee Han Nov 17 '16 at 03:53
  • The phrasing in the second question is either incorrect or misleading. "If $AB=0$ then neither $A$ nor $B$ can have an inverse" is false. The correct phrasing is "If $AB=0$ then either $A$ or $B$ doesn't have an inverse." It is possible for one of them to have an inverse but not both. For trivial example, $A=0, B=I$ you have $0I=0$ despite $I$ being invertible. – JMoravitz Nov 17 '16 at 04:21

2 Answers2

0

$A$ has an inverse iff there is $B$ where $AB=I.$ If $A=0,$ then no matter what $B$ is we get $AB=0,$ which is not $I.$ I leave the other question to you...

coffeemath
  • 29,884
  • 2
  • 31
  • 52
0

Suppose that $A$ and $B$ ae invertble. Then $0=A^{-1}ABB^{-1}=I$. Absurd !

Fred
  • 77,394