Relation between linear independence and matrix inversion

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My friend was wondering, suppose the invertible matrices $A_i$, $1 le i le k$, are linearly independent as vectors in $mathrmM_n times n(mathbbR)$. Is it true that the $A_i^-1$ are linearly independent if $k ge 3$?




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    My friend was wondering, suppose the invertible matrices $A_i$, $1 le i le k$, are linearly independent as vectors in $mathrmM_n times n(mathbbR)$. Is it true that the $A_i^-1$ are linearly independent if $k ge 3$?




    linear-algebra matrices




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      My friend was wondering, suppose the invertible matrices $A_i$, $1 le i le k$, are linearly independent as vectors in $mathrmM_n times n(mathbbR)$. Is it true that the $A_i^-1$ are linearly independent if $k ge 3$?




      linear-algebra matrices




      share|cite|improve this question











      My friend was wondering, suppose the invertible matrices $A_i$, $1 le i le k$, are linearly independent as vectors in $mathrmM_n times n(mathbbR)$. Is it true that the $A_i^-1$ are linearly independent if $k ge 3$?





      linear-algebra matrices





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      asked Jul 19 at 17:21









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          No. Take$$A_1=beginpmatrix1&0&0\0&frac12&0\0&0&frac13endpmatrix, A_2=beginpmatrixfrac12&0&0\0&frac13&0\0&0&frac14endpmatrix,text and A_3=operatornameId_3.$$They are linearly independent, but their inverses are not.






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            No. Take$$A_1=beginpmatrix1&0&0\0&frac12&0\0&0&frac13endpmatrix, A_2=beginpmatrixfrac12&0&0\0&frac13&0\0&0&frac14endpmatrix,text and A_3=operatornameId_3.$$They are linearly independent, but their inverses are not.






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              up vote
              1
              down vote



              accepted










              No. Take$$A_1=beginpmatrix1&0&0\0&frac12&0\0&0&frac13endpmatrix, A_2=beginpmatrixfrac12&0&0\0&frac13&0\0&0&frac14endpmatrix,text and A_3=operatornameId_3.$$They are linearly independent, but their inverses are not.






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                up vote
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                accepted







                up vote
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                down vote



                accepted






                No. Take$$A_1=beginpmatrix1&0&0\0&frac12&0\0&0&frac13endpmatrix, A_2=beginpmatrixfrac12&0&0\0&frac13&0\0&0&frac14endpmatrix,text and A_3=operatornameId_3.$$They are linearly independent, but their inverses are not.






                share|cite|improve this answer













                No. Take$$A_1=beginpmatrix1&0&0\0&frac12&0\0&0&frac13endpmatrix, A_2=beginpmatrixfrac12&0&0\0&frac13&0\0&0&frac14endpmatrix,text and A_3=operatornameId_3.$$They are linearly independent, but their inverses are not.







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                answered Jul 19 at 17:40









                José Carlos Santos

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