Question

Let and . Verify that

Answer 1

The given matrices are A=[ 3 7 2 5 ] and B=[ 6 8 7 9 ].

The value of AB is,

AB=[ 3 7 2 5 ][ 6 8 7 9 ] =[ 18+49 24+63 12+35 16+45 ] =[ 67 87 47 61 ]

The co-factors of elements of matrix AB are,

A 11 = ( −1 ) 1+1 61 =61

A 12 = ( −1 ) 1+2 47 =−47

A 21 = ( −1 ) 2+1 87 =−87

A 22 = ( −1 ) 2+2 67 =67

The adjoint of AB will be,

adj( AB )=[ A 11 A 21 A 12 A 22 ] =[ 61 −87 −47 67 ]

The determinant of AB is,

| AB |=67×61−( −87 )×( −47 ) =−2

Since,

( AB ) −1 = 1 | AB | adj( AB )

Substitute [ 61 −87 −47 67 ] for adjAB and −2 for | AB | in above formula,

( AB ) −1 =− 1 2 [ 61 −87 −47 67 ]

Now, the determinant of A is,

| A |=3×5−2×7 =1

And, the determinant of B is,

| B |=6×9−7×8 =−2

The co-factors of elements of matrix A are,

A 11 = ( −1 ) 1+1 5 =5

A 12 = ( −1 ) 1+2 2 =−2

A 21 = ( −1 ) 2+1 7 =−7

A 22 = ( −1 ) 2+2 3 =3

The adjoint of A will be,

adj( A )=[ A 11 A 21 A 12 A 22 ] =[ 5 −7 −2 3 ]

The co-factors of elements of matrix B are,

A 11 = ( −1 ) 1+1 9 =9

A 12 = ( −1 ) 1+2 7 =−7

A 21 = ( −1 ) 2+1 8 =−8

A 22 = ( −1 ) 2+2 6 =6

The adjoint of B will be,

adj( B )=[ A 11 A 21 A 12 A 22 ] =[ 9 −8 −7 6 ]

Thus,

A −1 =1[ 5 −7 −2 3 ]

B −1 =− 1 2 [ 9 −8 −7 6 ]

The value of B −1 A −1 is,

B −1 A −1 = −1 2 [ 9 −8 −7 6 ][ 5 −7 −2 3 ] = −1 2 [ 45+16 −63−24 −35−12 49+18 ] =− 1 2 [ 61 −87 −47 67 ]

Hence, ( AB ) −1 = B −1 A −1 is verified.