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

Algebra ->  Algebra  -> Matrices-and-determiminant -> 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      Log On

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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
matrix%281%2C3%2Cabs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29%2C+%27=%27%2C+-3%29++%29+
for x.
Found 2 solutions by jim_thompson5910, Edwin McCravy:
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
Take note that the matrix %28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29 is of the form
abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29 where a=4, b=x, c=-2, d=-x, e=-3, f=1, g=-6, h=2, and i=3


Since abs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29=-3, this means that d=-3



abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 Start with the general 3x3 determinant formula.



abs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29=%284%29%28%28-3%29%283%29-%281%29%282%29%29-%28x%29%28%28-x%29%283%29-%281%29%28-6%29%29%2B%28-2%29%28%28-x%29%282%29-%28-3%29%28-6%29%29 Plug in the given values (see above)


d=%284%29%28%28-3%29%283%29-%281%29%282%29%29-%28x%29%28%28-x%29%283%29-%281%29%28-6%29%29%2B%28-2%29%28%28-x%29%282%29-%28-3%29%28-6%29%29 Replace the left side with "d" (see above)


d=4%28-9-2%29-x%28-3x%2B6%29-2%28-2x-18%29 Multiply


d=4%28-11%29-x%28-3x%2B6%29-2%28-2x-18%29 Combine like terms.


d=-44-x%28-3x%2B6%29-2%28-2x-18%29 Multiply


d=-44%2B3x%5E2-6x%2B4x%2B36 Distribute


-3=-44%2B3x%5E2-6x%2B4x%2B36 Plug in d=-3


0=-44%2B3x%5E2-6x%2B4x%2B36%2B3 Add 3 to both sides.


0=3x%5E2-2x-5 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=3, b=-2, and c=-5


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%283%29%28-5%29+%29%29%2F%282%283%29%29 Plug in a=3, b=-2, and c=-5


x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%283%29%28-5%29+%29%29%2F%282%283%29%29 Negate -2 to get 2.


x+=+%282+%2B-+sqrt%28+4-4%283%29%28-5%29+%29%29%2F%282%283%29%29 Square -2 to get 4.


x+=+%282+%2B-+sqrt%28+4--60+%29%29%2F%282%283%29%29 Multiply 4%283%29%28-5%29 to get -60


x+=+%282+%2B-+sqrt%28+4%2B60+%29%29%2F%282%283%29%29 Rewrite sqrt%284--60%29 as sqrt%284%2B60%29


x+=+%282+%2B-+sqrt%28+64+%29%29%2F%282%283%29%29 Add 4 to 60 to get 64


x+=+%282+%2B-+sqrt%28+64+%29%29%2F%286%29 Multiply 2 and 3 to get 6.


x+=+%282+%2B-+8%29%2F%286%29 Take the square root of 64 to get 8.


x+=+%282+%2B+8%29%2F%286%29 or x+=+%282+-+8%29%2F%286%29 Break up the expression.


x+=+%2810%29%2F%286%29 or x+=++%28-6%29%2F%286%29 Combine like terms.


x+=+5%2F3 or x+=+-1 Simplify.


So the answers are x+=+5%2F3 or x+=+-1



Answer by Edwin McCravy(6938) About Me  (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:

matrix%281%2C3%2Cabs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29%2C+%27=%27%2C+-3%29++%29+

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

Suppose you have the 3x3 determinant:

abs%28matrix%283%2C3%2CA%2CB%2CC%2CD%2CE%2CF%2CG%2CH%2CI%29%29

Copy the first two columns over on the right:

abs%28matrix%283%2C3%2CA%2CB%2CC%2CD%2CE%2CF%2CG%2CH%2CI%29%29matrix%283%2C2%2CA%2CB%2CD%2CE%2CG%2CH%29

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 

matrix%281%2C3%2Cabs%28matrix%283%2C3%2CA%2CB%2CC%2CD%2CE%2CF%2CG%2CH%2CI%29%29matrix%283%2C2%2CA%2CB%2CD%2CE%2CG%2CH%29%2C+%27=%27%2C+AEI%2BBFG%2BCDH-CEG-AFH-BDI%29%29+


matrix%281%2C3%2Cabs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29%2C+%27=%27%2C+-3%29++%29+

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:

matrix%281%2C3%2Cabs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29matrix%283%2C2%2C4%2Cx%2C-x%2C-3%2C-6%2C2%29%2C+%27=%27%2C+-3%29++%29+

Then

matrix%289%2C3%2C%0D%0A%0D%0A%284%29%28-3%29%283%29%2B%28x%29%281%29%28-6%29%2B%28-2%29%28-x%29%282%29-%28-2%29%28-3%29%28-6%29-%284%29%281%29%282%29-%28x%29%28-x%29%283%29%2C+%27=%27%2C-3%2C%0D%0A-36-6x%2B4x%2B36-8%2B3x%5E2%2C%27=%27%2C+-3%2C%0D%0A-2x-8%2B3x%5E2%2C%27=%27%2C-3%2C%0D%0A3x%5E2-2x-8%2C%27=%27%2C-3%2C%0D%0A3x%5E2-2x-5%2C%27=%27%2C0%2C%0D%0A%283x-5%29%28x%2B1%29%2C%27=%27%2C0%2C%0D%0A3x-5=0%2Cx%2B1=0%2C%27%27%2C%0D%0A3x=5%2Cx=-1%2C%27%27%2C%0D%0Ax=5%2F3%2C%27%27%2C%27%27%29+

So there are two solutions, x=5%2F3 and x=-1

Edwin