5

How to show that a square matrix is not invertible if at least one row or column is zero ? I can show if a row is zero, the result C of $AB=C$ can not be the identity matrix because there is a zero row. But for the column case ?

Assume I don't know something about determinants.

fast-forward
  • 311
  • 2
  • 3
  • 9
  • 1
    Have you ever shown that an invertible matrix (over a field) has a two sided inverse? That is, if $A$ is invertible matrix with inverse $A^{-1}$, then $A^{-1}A = I = AA^{-1}$? – Alex Wertheim Oct 12 '13 at 07:36
  • It shouldn't be too bad trying to prove the column case in a way similar to what you did for rows. – Dennis Meng Oct 12 '13 at 07:38

2 Answers2

5

Hint :

Let $A$ be a square matrix such that $i^{th}$ column is zero.

For any $B\in M_{n\times n}$ what would be the $i^{th}$ column of $BA$?

  • i'm stupid, i've only looked at the $AB$ case. Therefore it takes a proof by contradiction – fast-forward Oct 13 '13 at 09:08
  • where is "proof by contradiction"?? you are not assuming anything.... you are just seeing what $BA$ looks like for any $B$.. anyways I guess you understood the point... –  Oct 13 '13 at 09:11
  • i am a bloody beginner in mathematical proofs but i thought i assume a square matrix will be invertible if at least one row or column is zero and showing that the assumption will be false!? this is the first exercise of a course – fast-forward Oct 13 '13 at 09:32
  • it is alright.. I was in a more worse condition than yours just before 2 months... It will all get set... good luck! –  Oct 13 '13 at 09:35
1

hint: theorem. let A be square invertible matrix. then [A,I] can be transformed into [I,A(inverse)] using elementary row operations.

but since A has a zero row or column, you can never transform the ith row or jth column be equal to 1 or that there will always be a zero in the diagonal which is not the identity matrix since [I]= 0 if i is not equal to j and 1 if i=j.

user116874
  • 11