Finding a 3x3 Matrix D that Satisfies a Given Equation

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The discussion revolves around finding a 3x3 matrix D that satisfies the equation ADA^{-1} = [[1,0,0],[1,0,0],[1,0,0]], given the matrix A and its inverse A^{-1}. The user expresses confusion about how to manipulate the equation to isolate D, particularly regarding the order of matrix multiplication. A suggested approach involves multiplying both sides of the equation by A^{-1} on the left and A on the right, which leads to the expression D = A^{-1}[[1,0,0],[1,0,0],[1,0,0]]A. The user acknowledges their misunderstanding and seeks clarification on the steps involved in this matrix operation. The conversation highlights the importance of understanding matrix multiplication order and properties in solving such problems.
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Hello everyone the following problem has me completely stumped, I am to find a certain 3x3 matrix D that satisfies the following equation:

ADA^{-1} = \left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)

where :

A = \left(\begin{array}{ccc}1&2&3\\0&1&1\\0&2&1\end{array}\right)

A^{-1} = \left(\begin{array}{ccc}1&-4&1\\0&-1&1\\0&2&-1\end{array}\right)

Heres my reasoning (or lack thereof), I know that AA^{-1} will yield the identity matrix I3, however clearly the D I am looking for is WITHIN this operation, and by matrix multiplication i cannot use this fact since the order is now completely different. But what I do know is how to find the inverse of A, but what property can I use for finding a 3x3 matrix? You see this would be simpler if they were happening to look for a 3x1 matrix D where I could use row operations in gauss jordan elimination to solve for the particular values, however I did not find any examples of this problem in the book--- where I am given an unknown nxn matrix to find and a certain operation that it must adhere to.

I could i solve this one? I have been understanding everything up to this point but i am clearly not understanding some simple rule--- thanks a lot for your help.
 
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Multiply both sides on the left by A inverse, then multiply both sides on the right by A...then D is on the left and you can expand out the other to find what D is.
 
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Wait what do you mean by expanding out the other? What does both sides on the left mean? Like A and D? Thanks.
 
btw i really apologize for the stupid thread title... i was ctually testing out my TeX format and accidentally posted with a wrong name--- id change it if i could but i cannot!
 
lol...its ok :)

EDIT: changed to tex

ADA^{-1}=\left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)

A^{-1}\left(ADA^{-1}=\left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)\right)

A^{-1}ADA^{-1}=A^{-1}\left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)

DA^{-1}=A^{-1}\left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)

\left(DA^{-1}=A^{-1}\left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)\right)A

DA^{-1}A=A^{-1}\left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)A

D=A^{-1}\left(\begin{array}{ccc}1&0&0\\1&0&0\\1&0&0\end{array}\right)A
 
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