Proving Singularity of Matrix B with Added Column Ab

[EV33]

1. Homework Statement
Let A = [A1,...,An-1] be an (nx(n-1)) matrix. Show that B = [A1,...,An-1,Ab] is singular for every choice of b in R^n-1.



2. Homework Equations
Ax = 0



3. The Attempt at a Solution
I know that if B is singular that means that for the equation Bx = 0 there exists another solution another than the trivial solution (x = 0). Now if we made B have all the same columns as A except added a new column Ab, that would make B a square matrix that is (nxn). But from there, I can't figure out how to use the information I know to solve the problem...

 

[radou]

If a matrix is singular, then its columns are linearly dependent. Any ideas?

 

[Matthollyw00d]

Ab=[A1b A2b ... An-1b]^T so by construction the column Ab is a linear combination of the first n-1 column vectors, regardless of what the vector b actually is. Hence detB=0

 

[EV33]

radou isn't that only true if the matrix is two by two?


And Matthollywood I am not sure what you are saying, could you please reword what you said.

Thank you.