# Thread: The Inverse of a Matrix

1. ## The Inverse of a Matrix

Hi all! I have two questions for y'all; talkin' bout Linear Algebra.

1) Explain why the columns of an n * n matrix A are linearly independent when A is invertible.

2) Suppose A in x * n and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to I(sub)n. This shows that A must be invertible.

Thanks sooo much guys!!! I really appreciate it! Linear Lolgebra ftl.

2. Originally Posted by WhoDatGrrl
Hi all! I have two questions for y'all; talkin' bout Linear Algebra.

1) Explain why the columns of an n * n matrix A are linearly independent when A is invertible.

2) Suppose A in x * n and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to I(sub)n. This shows that A must be invertible.

Thanks sooo much guys!!! I really appreciate it! Linear Lolgebra ftl.
Your $n \times n$ matrix is square and is most likely invertible. But generally a finite set with more than n vectors in $R^n$ must be linearly dependent because the homogeneous linear system whose coefficient matrix has those vectors as columns has more unknowns than equations and hence has nontrivial solutions.

Keep in mind that all of the three following statements are equivalent

° A is invertible.

° A homogeneous linear system Ax=0 has only the trivial solution

° The column and row vectors of A are linearly independent (because a matrix is invertible iff its transpose is invertible).