Can a Matrix with Identical Columns be Invertible?

[Chadlee88]

i need to be able to prove that an nxn matrix with two identical columns cannot be invertible. I know that if the columns of the matrix are linearly independent then the matrix is invertible. Could some please give me a hint on how to do this proof because i really don't know where to start.

 

[Galileo]

Depending on what has been covered in your class, it may be easiest to work with the nullspace of the matrix A. A is invertible iff the nullspace of A contains only the zero vector.
Call, the matrix A. If you can find a nonzero vector x such that Ax=0, then you've shown A is not invertible.

 

[HallsofIvy]

As Galileo said, it depends on what has been covered. It is relatively easy to show that the determinant of a matrix in which two columns are the same is 0.

 

[Pythagorean]

i need to be able to prove that an nxn matrix with two identical columns cannot be invertible. I know that if the columns of the matrix are linearly independent then the matrix is invertible. Could some please give me a hint on how to do this proof because i really don't know where to start.

what's it mean when two columns of a matrix are identical?

Compare this with what is meant by 'linearly independent'.

Can two identical columns in one matrix be independent?

 

[HallsofIvy]

Actually, I've never seen a text that definedf "independent" for the columns of a matrix!

You can, of course, think of the columns of a matrix as vectors and then determine whether or not those vectors are independent.

 

[Office_Shredder]

hallsofivy, I think the columns being independent refers to them as vectors being linearly independent essentially (it's not proper terminology perhaps, but it does get the point across)

 

[HallsofIvy]

hallsofivy, I think the columns being independent refers to them as vectors being linearly independent essentially (it's not proper terminology perhaps, but it does get the point across)

Yeh, I went back and edited my post just before I saw this.