Question

Choose the correct answer. If x , y , z are nonzero real numbers, then the inverse of matrix is A. B. C. D.

Answer 1

The determinant is given as,

A=| x 0 0 0 y 0 0 0 z |

The inverse of matrix A can be calculated as,

A −1 = 1 | A | adj( A ), it exists if | A |≠0.

Now, calculate | A |.

| A |=| x 0 0 0 y 0 0 0 z | =x( yz−0 )−0( 0−0 )+0( 0−0 ) =xyz

Here, | A |≠0, so the inverse of A exists.

Further we have to calculate adj A,

A=[ B 11 B 21 B 31 B 12 B 22 B 32 B 13 B 23 B 33 ]

The minors can be calculated as,

C 11 =| y 0 0 z |=yz C 12 =| 0 0 0 z |=0 C 13 =| 0 y 0 0 |=0 C 21 =| 0 0 0 z |=0

Further calculate the minors as,

C 22 =| x 0 0 z |=xz C 23 =| x 0 0 0 |=0 C 31 =| 0 0 y 0 |=0 C 32 =| x 0 0 0 |=0

Further calculate the minors as,

C 33 =| x 0 0 y |=xy

Calculate the cofactors as,

B 11 = ( −1 ) 1+1 C 11 = ( −1 ) 2 ×yz=yz B 12 = ( −1 ) 1+2 C 12 = ( −1 ) 3 ×( 0 )=0 B 13 = ( −1 ) 1+3 C 13 = ( −1 ) 4 ×( 0 )=0 B 21 = ( −1 ) 2+1 C 21 = ( −1 ) 3 ×( 0 )=0

Further calculate the factors as,

B 22 = ( −1 ) 2+2 C 22 = ( −1 ) 4 ×xz=xz B 23 = ( −1 ) 2+3 C 23 = ( −1 ) 5 ×( 0 )=0 B 31 = ( −1 ) 3+1 C 31 = ( −1 ) 4 ×( 0 )=0 B 32 = ( −1 ) 3+2 C 32 = ( −1 ) 5 ×( 0 )=0

Further calculate the factors as,

B 33 = ( −1 ) 3+3 C 33 = ( −1 ) 6 ×xy=xy

By substituting all value in the matrix A we get,

adj( A )=[ yz 0 0 0 xz 0 0 0 xy ]

By substituting the value of adj( A ) in the formula of inverse of matrix A we get,

A −1 = 1 xyz [ yz 0 0 0 xz 0 0 0 xy ] =[ x −1 0 0 0 y −1 0 0 0 z −1 ]

Thus, the correct option is (A).