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monroe17
 2 years ago
use cofactor expansion to solve for the determinant of the matrix
5 11 8 7
3 2 6 23
0 0 0 3
0 4 0 17
monroe17
 2 years ago
use cofactor expansion to solve for the determinant of the matrix 5 11 8 7 3 2 6 23 0 0 0 3 0 4 0 17

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SithsAndGiggles
 2 years ago
Best ResponseYou've already chosen the best response.0\[\left\begin{matrix}5&11&8&7\\ 3&2&6&23\\ 0&0&0&3\\ 0&4&0&17\end{matrix}\right\] Using a cofactor expansion will be significantly simpler if you use the third row. So, the expansion is \[\left\begin{matrix}5&11&8&7\\ 3&2&6&23\\ \color{red}0&\color{red}0&\color{red}0&\color{red}{3}\\ 0&4&0&17\end{matrix}\right=0C_{3,1}+0C_{3,2}+0C_{3,3}+(3)C_{3,4}\] where C_{i,j} is the cofactor of a_{i,j}, and the cofactor is \[C_{i,j}=(1)^{i+j}M_{i,j},\] where M_{i,j} is the minor of a_{i,j}. I hope none of this is new material. Right away, you can simplify the expansion to \[\begin{align*}\left\begin{matrix}5&11&8&7\\ 3&2&6&23\\ \color{red}0&\color{red}0&\color{red}0&\color{red}{3}\\ 0&4&0&17\end{matrix}\right&=(3)(1)^{3+4}\left\begin{matrix}5&11&8\\3&2&6\\0&4&0\end{matrix}\right\\ &=3\left\begin{matrix}5&11&8\\3&2&6\\0&4&0\end{matrix}\right \end{align*}\] Do you think you can take it from here? I'd suggest another expansion using the third row.

monroe17
 2 years ago
Best ResponseYou've already chosen the best response.0it is new material ;/but i sorta understood what you just did.. but what do i do with that 3? enable to expand again?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2the 4x4 was effectively reduced to a 3x3, which is much simpler to play with. you can use that last row that is full of zeros to reduce it again

monroe17
 2 years ago
Best ResponseYou've already chosen the best response.0so starting off where @SithsAndGiggles left off.. would i do (3)(4)* [1^{3+2}] of 5 11 3 2

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2the matrix we have to work with is: 5 11 8 3 2 6 0 4 0 lets cross out the row and colum with the 4 in it 5  8 3  6    whats left is the matrix we want to attach to the 4 part; but there is a ++ to sort out as well

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2+  +  +  +  + notice that this sign board has a  in the place where the 4 would be; that tells us that we want (4) as the scalar so it looks like you had it worked our correctly other than the submatrix parts :)

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2why do i see a 4 in there? now that i reread it lol, the sign board has a  where the 4 is; giving us a scalar of (4)

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.23 * (4) * 5 8 3 6 and of course the determinant of that 2x2 is 5(6)3(8)

monroe17
 2 years ago
Best ResponseYou've already chosen the best response.0wait what happened to.. (1)^3+2?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2thats another way to represent the sign board; I never remember the power stuff so I just work out it out with the sign board

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2top left corner is always a +, and it checkerboards from there

monroe17
 2 years ago
Best ResponseYou've already chosen the best response.0oh okay and then I just carry that 3 from the 1st expansion along the ride lol till the end.

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2yep, its stuck on there luck .... well something thats stuck lol

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2your reducing each matrix to a simpler form, the scalars from the previous attempts dont vanish

monroe17
 2 years ago
Best ResponseYou've already chosen the best response.0can you explain the "sign board" and little more clealy ? :) everything else makes sense

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2the sign board is a term im calling it for lack of a better term :) the manner in which a determinant is worked out ... culling the parts from the matrix ....results in either adding or subtracting certain amounts. the sign board is a visual account of not shuffling the pieces around really.

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2a 2x2 is the simplest case to use for an example; why do we have a subtraction sign in there?

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2take the example: 0 4 3 0 the determinant of this using submatrix parts is; lets use the 4 and cancel out the row and column its in   3 this gives us 4(3) as the determinant if we dont adjust for signs but we know that the actual determinant is 0(0)  4(3) = 12 the mechanics of the matrix present us with this sign board such that for every scalar we want to use we have to adjust for its placement: +   + we used the scalar for the placement if the 4, and its submatrix (3) giving us a 4(3) determinant

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.2when the matrixes get large, the sign board is equivalent to using the (1)^(n+m) adjustment :)

monroe17
 2 years ago
Best ResponseYou've already chosen the best response.0that's why that was used! lol that confused me, but it made sense.
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