SOLUTION: Hello! How do I solve this determinant problem. I know the answer is 5/3 , -1 but am not sure how? Thanks for any help. Solve {{{matrix(1,3,abs(matrix(3,3,4,x,-2,-x,-3,1,-6

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Question 177030: Hello!
How do I solve this determinant problem.
I know the answer is 5/3 , -1 but am not sure how?
Thanks for any help.
Solve

for x.

Found 2 solutions by jim_thompson5910, Edwin McCravy:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Take note that the matrix is of the form
where , , , , , , , , and


Since , this means that



Start with the general 3x3 determinant formula.



Plug in the given values (see above)


Replace the left side with "d" (see above)


Multiply


Combine like terms.


Multiply


Distribute


Plug in


Add 3 to both sides.


Combine like terms.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the answers are or



Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!

Edwin's solution follows. You do not need the quadratic formula, as the resulting quadratic factors:



First you have to learn how to find the value of
a 3x3 determinant:

Suppose you have the 3x3 determinant:



Copy the first two columns over on the right:



Then add the products of the three diagonals that go down to the
RIGHT:  +AEI+BFG+CDH

Then subtract the products of the three diagonals that go down to the
LEFT:  -CEG-AFH-BDI

So 






Evaluate the determinant according to the rules 
above or whatever rules you learned. There are more
than one possible set of rules for this.  Use 
whatever method you have studied.  If you use the 
method above, you'll copy the first two columns over:



Then



So there are two solutions,  and 

Edwin

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