How Does Proving A Matrix Is Row Equivalent to the Identity Matrix Work?

[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'm CLUELESS as to WHERE TO START. Please help me

I tried simplifying to the matrix

[1 d]
[0 b - cd]

And have no clue what to do next.

 

[daveb]

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

 

[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?

 

[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.