# Math Help - Cramer Rule and Vector Spaces

1. ## Cramer Rule and Vector Spaces

1) How do you find A (by hand) if an adjoint of A 4 by 4 matrix is given you?

For example:

2 0 0 0
0 2 1 0
0 4 3 2
0 -2 -1 2

what is A?

I know that A (adj. A) = (det. A) I
and I tried to find A by finding the Adj. A of the given Adj. of A and I got the weird numbers.

The answer is suposed to give the following matrix:
A equals
1 0 0 0
0 4 -1 1
0 -6 2 -2
0 1 0 1

How do you get to that? I had asked this several days ago and the links I received sort of helped but I got confused again. I tried to solve this twice and its frustrating that I am not getting the answer. Please if anyone could explain this with baby steps I don't mind at all I just want is to understand it. Thanks

2) Let R denote the set of real numbers. Define scalar multiplication by
a x= a *x (the usual multiplication of real numbers)
and define the addition, denoted Å , by
x Å y = max (x, y) (the maximum of the two numbers)
( I have no idea how to solve this one either)

2. 1) If A is an n by n matrix then det(adj A) = (det A)^(n-1). Also, if det A is not zero then A = (det A)(adj A)^(-1). For this example,

det(adj A) $=\begin{vmatrix}2&0&0&0\\0&2&1&0\\0&4&3&2\\0&-2&-1&2\end{vmatrix} = 8 = 2^3$, and so det A = 2. You then have to find the inverse of adj A and multiply it by 2, in order to get A.

2) A vector space has to have a zero element. If R with the given operation is a vector space then there must exist an element O of R such that x Å O = x for every x in R. Think about what that says about O. (Of course, O need not be the usual 0. In fact, it would have to have very unusual properties.)

3. Originally Posted by googoogaga
2) Let R denote the set of real numbers. Define scalar multiplication by
a x= a *x (the usual multiplication of real numbers)
and define the addition, denoted Å , by
x Å y = max (x, y) (the maximum of the two numbers)
( I have no idea how to solve this one either)
You go through the list of axioms defining a vector space showing that Å
does or does not satisfy these axioms over the reals.

The one you will have a problem with the one Opalg pointed out, in fact you
will still have problems if you work with extended reals as you will need

+infty+(-infty)=0,

but this combination of the two infinities is usually left undefined.

RonL

4. sorry opalg I thought I undestood the first problemm. I got stuck again. How do you find the inverse of the given adjoint of A I am lost completely eveything you wrote made sense until I started working on it again. How do you get to the matrix I have to see it in order to understand it. I thought I did with the explanations you gave me but I am getting nowhere. Please help...

5. Originally Posted by googoogaga
How do you find the inverse of the given adjoint of A
Same way as you would find the inverse of any other matrix. And that depends on which method you normally use for finding inverses. Since this is a problem about adjoint matrices, maybe you know the formula $A^{-1} = \frac{\mathrm{adj}(A)}{\mathrm{det}(A)}$? In that case, you can find the inverse of adj(A) by calculating adj(adj(A)). In fact, $(\mathrm{adj}(A))^{-1} = \frac{\mathrm{adj}(\mathrm{adj}(A))}{\mathrm{det}( \mathrm{adj}(A))}$.