few things about determinants & matrix

[swaha]

1. prove that the value of a determinant is unique? ie the ans is same whichever way we expand it starting from whichever column?

2. det(AB)=det(A)det(B) why? where A,B r determinants of order n?

3. for a matrix to be orthogonal (A transpose)(A)=(A)(A inverse).

again if AB=AC we cant say that B=C. so how can we say in the previous case that A transpose=A inverse?

4. is identity matrix unique can be proved? ie if AB=A can we say B is unique? if yes how?

 

 

thanks in advance.

[ajb]

These questions don't seem particularly hard. Pick one and show us how far you have got already. I doubt anyone will provide full proofs of your questions without you having shown some effort.

[swaha]

i can prove all them with 2 by 2 & 3 by determinants.

1. this would also be true if no signs needed to be considered. dont know how to proceed wid signs.

2. no idea for this one.

3, 4 i did them, so not reqd any more.

got another prob if the elements of a row be multiplied wid cofacors corr to some other row th result vanishes. cant prove it beyond 3 by 3.& dont know how to apply induction here.

[ajb]

I'd be tempted to use index notation and the Levi-Civita symbol to write out determinants of a n by n matrix (n finite).