If A^2 = 0, then A is not an invertible matrix

  • #1
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Homework Statement


Suppose that ##A^2 = 0##. Show that ##A## is not an invertible matrix

Homework Equations




The Attempt at a Solution


We can do a proof by contradiction. Assume that ##A^2 = 0## and that ##A## is invertible. This would imply that ##A=0##, which is to say that A is not invertible, since ##0## has no inverse. This is a contraction, so it must be the case that if ##A^2 = 0##, then ##A## is not invertible.

Is this the way I should be doing this problem?
 

Answers and Replies

  • #2
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Homework Statement


Suppose that ##A^2 = 0##. Show that ##A## is not an invertible matrix

Homework Equations




The Attempt at a Solution


We can do a proof by contradiction. Assume that ##A^2 = 0## and that ##A## is invertible. This would imply that ##A=0##, which is to say that A is not invertible, since ##0## has no inverse. This is a contraction, so it must be the case that if ##A^2 = 0##, then ##A## is not invertible.

Is this the way I should be doing this problem?
I first thought - and this might well have been intended - that you should show, that there is a non-trivial element in the kernel of ##A##, namely the entire image of ##A##, but I like your solution better.
 
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Likes Mr Davis 97
  • #3
Math_QED
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Homework Statement


Suppose that ##A^2 = 0##. Show that ##A## is not an invertible matrix

Homework Equations




The Attempt at a Solution


We can do a proof by contradiction. Assume that ##A^2 = 0## and that ##A## is invertible. This would imply that ##A=0##, which is to say that A is not invertible, since ##0## has no inverse. This is a contraction, so it must be the case that if ##A^2 = 0##, then ##A## is not invertible.

Is this the way I should be doing this problem?
Looks good to me as well. Note that you can prove this directly by using determinants, but I suspect you are not allowed to use determinants at this stage.
 
  • #4
ehild
Homework Helper
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We can do a proof by contradiction. Assume that ##A^2 = 0## and that ##A## is invertible. This would imply that ##A=0##,
There are nonzero matrices so as A2=0. You should prove that they are not invertible.
For example, the square of the following matrix is zero.
\begin{pmatrix}

0 & 1 \\
0 & 0

\end{pmatrix}
 
Last edited:
  • #5
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There are nonzero matrices so as A2=0. You should prove that they are not invertible.
If A^2 = 0 and A is invertible, this implies A^(-1) A^2 = A^(-1) 0 = 0. No need to bother with non-invertible A's here.
 

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