# Is an invertible matrix always lin. independent?

1. May 14, 2009

### hydralisks

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?

2. May 14, 2009

### Pengwuino

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

3. May 14, 2009

### HallsofIvy

Staff Emeritus
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.