यदि `x,y,z` शून्येतर वास्तविक संख्याएं हों तो आव्यूह `A=[(x,0,0),(0,y,0),(0,0,z)]` का व्युत्क्रम है:

Options

A. `[(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1))]`
B. `xyz[(x^(-1),0,0),(x,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)(0,0,0z)]`
`implies|A|=|(x,0,0),(0,y,0),(0,0,z)|=xyz!=0`
`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_(33)=xy`
`:.adjA=[(yz,0,0),(0,zx,0),(0,0,xy)]'=[(yz,0,0),(0,zx,0),(0,0,xy)]`
तथा `A^(-1)=1/(|A|).adjA=1/(xyz)[(yz,0,0),(0,zx,0),(0,0,xy)]`
`=[(1/x,0,0),(0,1/y,0),(0,0,1/2)]=[(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1))]`