Choose the correct answer in questions 17 to 19:
If x, y, z are nonzero real numbers then the inverse of metrix `A=[{:(x,0,0),(0,y,0),(0,0,z):}]`is :
(a) `[{:(x^(-1),0,0),(0,y^(1),0),(0,0,z^(1)):}]`
(b) `xyz[{:(x^(-1),0,0),(0,y^(1),0),(0,0,z^(1)):}]`
( c) `1/(xyz)[{:(x,0,0),(0,y,0),(0,0,z):}]`
(d) `1/(xyz)[{:(1,0,0),(0,1,0),(0,0,1):}]`

Answer

(a)

Explanation

`:." "A=[{:(x,0,0),(0,y,0),(0,0,z):}]`
`rArr" "|A|=[{:(x,0,0),(0,y,0),(0,0,z):}]=xyzne0`
`A_(11)=yz, A_(12)=0, A_(13)=0`
`A_(21)=0, A_(22)=zx, A_(23)=0`
`A_(31)=0, A_(32)=0, A_(32)=0, A_(33)=xy`
`:." adj A"=[{:(yz,0,0),(0,zx,0),(0,0,xy):}]=[{:(yz,0,0),(0,zx,0),(0,0,xy):}]`
`"and A"^(-1)=1/|A|".adj A"=1/(xyz)[{:(yz,0,0),(0,zx,0),(0,0,xy):}]`
`=[{:(1/x,0,0),(0,1/y,0),(0,0,1/z):}]`
`=[{:(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1)):}]`