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I wonder how can I prove

A matrix $A_(mxn)$ with $m \lt n$ has no left inverse and a matrix $A_(mxn)$ with $m \gt n$ has no right inverse

Because I got no idea about that

onurctirtir
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  • Do you know a theorem about the rank of a product of matrices? – hmakholm left over Monica Apr 10 '16 at 17:53
  • Hello, Onur. Please see this post about how to ask for help with homework. Also, excellent advice on how to ask a good question in general can be found here. – AnonSubmitter85 Apr 10 '16 at 18:03
  • It is also worth mentioning for typesetting, X_lotsofstuff displays as $X_lotsofstuff$ whereas X_{lotsofstuff} displays as $X_{lotsofstuff}$. If you have more than one character in a subscript or superscript, enclose it in braces. – JMoravitz Apr 10 '16 at 18:17

1 Answers1

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Hints:

  • $rank(A_{m\times n})\leq \min(m,n)$

Immediate from interpretation of rank as number of pivots in RREF form of matrix

  • $rank(AB)\leq \min(rank(A),rank(B))$

See this question

  • $rank(I_{n\times n})=n$

Immediate from interpretation of rank as number of pivots in RREF form of matrix

Further hint, if it so happened that $A$ has a left inverse, $B$, and $m<n$, in other words $BA=I$, what shape is $BA$? How many rows, how many columns? What is the rank of $A$? What is the rank of $I$?

Community
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JMoravitz
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