Question

Verify A ( adj A ) = ( adj A ) A = I .

Answer 1

Given matrix is,

A=[ 2 3 −4 −6 ]

The determinant of the given matrix is,

| A |=2×( −6 )−( −4 )×3 =−12+12 =0

Calculate | A |I,

| A |I=0×[ 1 0 0 1 ] =[ 0 0 0 0 ]

Cofactor of a 11 is,

A 11 = ( −1 ) 1+1 ( −6 ) =−6

Cofactor of a 12 is,

A 12 = ( −1 ) 1+2 ( −4 ) =4

Cofactor of a 21 is,

A 21 = ( −1 ) 2+1 ( 3 ) =−3

Cofactor of a 22 is,

A 22 = ( −1 ) 2+2 ( 2 ) =2

So the adjoint of the given matrix is,

adjA=[ A 11 A 21 A 12 A 22 ] =[ −6 −3 4 2 ]

Calculate A( adjA ),

A( adjA )=[ 2 3 −4 −6 ][ −6 −3 4 2 ] =[ −12+12 −6+6 24−24 12−12 ] =[ 0 0 0 0 ] =| A |I

Calculate ( adjA )A,

adjA( A )=[ −6 −3 4 2 ][ 2 3 −4 −6 ] =[ −12+12 −18+18 8−8 12−12 ] =[ 0 0 0 0 ] =| A |I

Hence, it is verified that A( adjA )=adjA( A )=| A |I.