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Question

If x,y, z are non-zero real numbers, then the inverse of matrix

A=x000y000z is

a) x1000y1000z1

b) xyzx1000y1000z1

c) 1xyzx000y000z

d) 1xyz100010001

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Solution

Given, A=x000y000z
|A|=x000y000z=x(yz=0)=xyz0 ( x, y and z are non - zero )
Cofactors of A are
A11=(yz0)=yz,A12=(00)=0A13=00=0A21=(00)=0,A22=xz0=xzA23=(00)=0A31=00=0A32=(00)=0A33=(xy0)=xy
adj(A)=yz000xz000xy=yz000xz000xy
Now, A1=1|A|(adj A)
=1xyzyz000xz000xy=⎢ ⎢ ⎢1x0001y0001z⎥ ⎥ ⎥=x1000y1000z1
Hence, the correct option is(a)


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