Homework Help: Orthogonal vector proof

1. Dec 12, 2012

Clandry

Let A be an n × n invertible matrix. Show that if
i ≠ j, then row vector i of A and column vector
j of A-1 are orthogonal.

I'm lost in regards to where to lost.

I want to show that a vector from row vector i from A is orthogonal to a column vector j from A.
Orthogonal means the dot product is 0 or the angle between the 2 vectors is 90 degrees.

After stating the obvious I'm stuck. I think I need to start with figuring out what the relationship between a row vector from A and a column vector from A^-1 is, but how do I do that?

2. Dec 12, 2012

Michael Redei

How would you show that the dot product of (row vector i of A) and (column vector j of A-1) is equal to 0? And in what kind of operation would you perform several such calculations between row vectors of one matrix and column vectors of another?

3. Dec 12, 2012

Clandry

If v represents the row vector and w represents the column vector.
I must show that v*wt=0. * means dot product.

The dot product would be then computed by multiplying each element in v by each corresponding element in w and taking the sum of all those products. But I'm not sure how this will show what the problem wants.

4. Dec 12, 2012

Michael Redei

Apart from calculating dot products, when does one multiply the elements of a row with the elements of a column and take the sum of the results?

If that's no help, try this: how would you prove that A-1 really is an inverse of A? What kind of calculation could you perform to show this?

5. Dec 12, 2012

Clandry

a-1*a=i?

6. Dec 12, 2012

Michael Redei

How does one multiply one matrix by another? Say you wanted to multiply A by A-1. Can you explain how you'd obtain the element in row i and column j of the resulting matrix?

7. Dec 12, 2012

Clandry

OH! I see what you're saying. You'd multiply the row by the column and sum up the products.
This would explain how to multiply the 2 vectors together.

8. Dec 12, 2012

Michael Redei

All you need now is some criterion to decide which elements in AA-1 can be zero and which can't.

9. Dec 12, 2012

Clandry

Okay Thanks for the help.

I am still having trouble. Or maybe I understand it but just not realize it.

In AA-1=I, everything off the diagonal is 0. Everything on the diagonal is nonzero, but that's when i=j. Everything off the diagonal is 0 b/c i≠j.

Is that the idea?

10. Dec 13, 2012

Michael Redei

That's exactly the right idea. You can safely ignore the diagonal, since your initial question was only about i≠j.