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Is an invertible matrix always lin. independent?

  • Thread starter hydralisks
  • Start date
  • #1
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Because according to my book,

If A's an invertible nxn matrix, then the eq. Ax=b is consistent for EACH b...

this consistency would imply lin. independency?
 

Answers and Replies

  • #2
Pengwuino
Gold Member
4,989
15
Yes. Also, the determinant of an invertible matrix is non-zero. The determinant of a matrix of linearly dependent vectors is 0.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
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Strictly speaking, the term "independent" applies to sets of vectors, not matrices so it does not make sense to ask if a matrix is independent. What you are really asking is whether the rows or columns of the matrix, thought of as vectors are independent. The answer to that is yes. If a matrix is invertible, then its rows (or columns), thought of as a set of vectors, is indepependent.
 

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