Homework Help: Help on simple linear algebra problem

1. Apr 28, 2006

Anisotropic Galaxy

Let A and I be as follows.

A = [1 d]
[c b]
I=[1 0]
[0 1]

Prove that if b - cd != 0, then A is row equivalent to I

I tried simplifying to the matrix

[1 d]
[0 b - cd]

And have no clue what to do next.

2. Apr 28, 2006

daveb

What is the criteria for row equivalent matrices? How would you perform the allowable operations to get to I?

3. Apr 28, 2006

HallsofIvy

To get $\left(\begin{array}{cc}1 & 0\\0 & 1\end{array}\right)$
you will need a 1 in place of that b-cd. What row operation will give you that?

4. May 18, 2006

jdog

It has been a while since I took linear algebra so I forget the terms for these things, but I know what you're getting at. If A is row equivalent to I, that means that elementary row operations can reduce it to such. That can only be done if the determinant is not zero. (Then we say A is either singular or not singular, don’t remember which) The determinant of A is b - cd. So in a sense, you're done, unless you actually need to prove what I just said.

In that case, argue by contradiction. Show that if b - cd = 0, The reduced row echelon form of A is not the identity matrix.

Hope that helps.