# Is an invertible matrix always lin. independent?

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

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Pengwuino
Gold Member
Yes. Also, the determinant of an invertible matrix is non-zero. The determinant of a matrix of linearly dependent vectors is 0.

HallsofIvy
Science Advisor
Homework Helper
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.