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Question 549363: Invert the matrix by computing the adjoint matrix and the determinant:
-1 1 3
0 0 -1
2 0 1
The answer is supposed to be:
0 1/2 1/2
1 7/2 1/2
0 -1 0
I want to know the method please
Answer by mathie123(224)   (Show Source):
You can put this solution on YOUR website! (-1 1 3) (x11 x12 x13)=(1 0 0)
(0 0 -1) (x21 x22 x23)=(0 1 0)
(2 0 1) (x31 x32 x33)=(0 0 1)
From the definition above we want that middle matrix----(that's what an inverse is).
Let's first find the determinant of A which is
(-1 1 3)
(0 0 -1)
(2 0 1)
So DetA=-1*detB+0*detC-0*detD
Where B is
(0 -1)
(2 1)
and DetB=0*1-(-1*2)=2
C is
(-1 3)
(2 1)
and detC=-1*1-3*2=-1-6=-7
and D is
(-1 3)
(0 -1)
and detD=-1*-1+3*0=1
So DetA=-1*2=-2
Now we need to find AdjA
CoeffMatrix
=(+detE -detB +detF)
(-detG +detC -detH)
(+detI -detD +detJ)
=AdjA(transpose)
where B,C and D are the same as above,
E=(0 -1)
(0 1)
detE=0*1-(-1*0)=0
From here on, I will just write the determinants, they are calculated the same way we have been doing above
F=(0 0)
(2 0)
detF=0
G=(1 3)
(0 1)
detG=1
H=(-1 1)
(2 0)
detH=-2
I=(1 3)
(0 -1)
detI=-1
J=(-1 1)
(0 0)
detJ=0
Leaving
CoeffMatrix
=(0 -2 0)
(-1 -7 2)
(-1 -1 0)
And AdjA is the transpose of CoeffMatrix
So AdjA=
(0 -1 -1)
(-2 -7 -1)
(0 2 0)
So finally, to find our inverse we have:
InverseA=(1/detA)*AdjA
=(1/-2)*AdjA
=(0 1/2 1/2)
(1 7/2 1/2)
(0 -1 0)
Hopefully this helps a bit:)
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