# Homework Help: Linear Algebra help

1. Feb 21, 2012

### songofmisery

let A be a 3x3 matrix. Suppose that for every row vector y=[y1 y2 y3] there exists a row vector x=[x1 x2 x3] such that xA=y. Show that A is invertable

i honestly have no idea where to even go with this. any help would be appreciated (:

2. Feb 21, 2012

### jbunniii

Hint: A is invertible if and only if there is a 3x3 matrix B such that BA = ?

3. Feb 21, 2012

### songofmisery

such that BA = I?

i read something about that in my text book but i dont understand what I is. Is it just the inverse?

4. Feb 21, 2012

### songofmisery

i know how to find an inverse, i just dont understand the part with row vectors and where it fits into the equation

5. Feb 21, 2012

### jbunniii

No, I is the identity matrix:

$$I = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$$

6. Feb 21, 2012

### jbunniii

Try to write BA = I, one row at a time.

7. Feb 21, 2012

### songofmisery

oh, that would make sense.

so could i say that B is x and I is y?

i feel like thats completely wrong.

8. Feb 21, 2012

### jbunniii

No, B and I are 3x3 matrices, whereas x and y are 1x3.

If you're having trouble seeing what to do, I suggest naming the elements of the matrix B, for example as follows:

$$B = \left[ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]$$

Now, what's the first row of BA = I? It is of the form xA = y. What are x and y in this case?

9. Feb 21, 2012

### songofmisery

x would be [a b c] and y would be [1 0 0]?

10. Feb 21, 2012

### jbunniii

Right. So now, reverse the argument. You are given the fact that for every row vector y, there is a row vector x such that xA = y. So start by choosing y = [1 0 0], and writing the corresponding x = [a b c]. Now repeat for the remaining two rows. What do you end up with?