1. ## Matrix Identity

Hey all,

Given $\displaystyle $B = \left( {\begin{array}{ccc} 1 & 0 & 2 \\ 3 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} } \right)$$ and $\displaystyle $B^{-1} = \left( {\begin{array}{ccc} -1 & 0 & 2 \\ 3 & 1 & -6 \\ 1 & 0 & -1 \\ \end{array} } \right)$$
find C, where $\displaystyle BCB^{T} = I$
Thanks.

2. What ideas have you had so far?

3. $\displaystyle BCB^{T}=I \Rightarrow BB^{T}C = I \\$
so $\displaystyle C = I(BB^{T})^{-1}\\$
which is $\displaystyle $M = \left( {\begin{array}{ccc} 11 & 3 & -2 \\ 3 & 1 & -6 \\ -21 & -6 & 41 \\ \end{array} } \right)$$
That seems to work.. not sure if its right though...

4. You might just get the right answer for the wrong reasons. *slaps your wrist*

You don't know that $\displaystyle B$ and $\displaystyle C$ commute! You can't just slip $\displaystyle B^{T}$ past the $\displaystyle C$ like that. However, you can left- and right-multiply by inverses, if you know they exist. In this case, you've been given $\displaystyle B^{-1}$, so you know that exists. How about this first step:

$\displaystyle BCB^{T}=I$

$\displaystyle B^{-1}BCB^{T}=B^{-1}I=B^{-1}$

$\displaystyle CB^{T}=B^{-1}.$

What else could you do?

5. (In regards to my second post - Not ackbeets)
Sorry thats wrong... I think it should be that
The product of $\displaystyle BC$ should give the inverse of $\displaystyle B^{T}$
so $\displaystyle B \cdot C = B^{-1} \Rightarrow C = B^{T-1} \cdot B^{-1}$
Which is

$\displaystyle $M = \left[ {\begin{array}{ccc} 5 & -15 & -3 \\ -15 & 46 & 9 \\ -3 & 9 & 2 \\ \end{array} } \right]$$

1. What is $\displaystyle M?$ No need to introduce another variable if you don't need it.

2. The notation $\displaystyle B^{T-1}$ should really be $\displaystyle (B^{T})^{-1}.$ Otherwise, someone might think you're raising $\displaystyle B$ to the power $\displaystyle T-1.$

3. You've got the order wrong. You need to right-multiply by $\displaystyle (B^{T})^{-1}.$ That is, you've got to do this (picking up from where I left off in my last post):

$\displaystyle CB^{T}=B^{-1}$

$\displaystyle CB^{T}(B^{T})^{-1}=B^{-1}(B^{T})^{-1}$

$\displaystyle C=B^{-1}(B^{T})^{-1}.$

How can you compute the RHS here?

7. Ah, thanks for the pointers. RHS would just be the product of the inverses of B and its Transpose..

8. Ah, but you're not directly given $\displaystyle (B^{T})^{-1}.$ How can you get it?

9. Wouldn't that just be:

$\displaystyle $B = \left( {\begin{array}{ccc} 1 & 0 & 2 \\ 3 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} } \right)$$ Then $\displaystyle $(B^{T})^{-1} = \left( {\begin{array}{ccc} 1 & 3 & 1 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \\ \end{array} } \right)^{-1}$$

10. True, but I think you can get what you need faster than that, if you use an identity.