Question

Find the inverse of the matrix `[(1      2     3),(1    1     5),(2    4     7)]` by adjoint method

Answer 1

Let A = `[(1       2     3),(1     1     5),(2     4     7)]`

A11 = (-1)1+1	M11 = (-1)2 (7 - 20) = -13
A12 = (-1 )1+2	M12 = (-1)3 (7 - 10) = 3
A13 = (-1 )1+3	M13 = (-1)4 (4 - 2) = 2
A21 = (-1 )2+1	M21 = (-1)3 (14 - 12) = -2
A22 = (-1 )2+2	M22 = (-1)4 (7 - 6) = 1
A23 = (-1 )2+3	M23 = (-1)5 (4 - 4) = 0
A31 = (-1 ) 3+1	M31 = (-1)4 (10 - 3) = 7
A32 = (-1 )3+2	M32 = (-1)5 (5 - 3) = -2
A33 = (-1 )3+3	M33 = (-1)6 (1 - 2) = -1

∴ Matrix of cofactor = `[(-13,3,2),(-2,1,0),(7,-2, -1)]`

Adj A = Transpose of the cofactor matrix [c_ij]

i.e. Adj A = [c_ij]' = `[(-13,-2,7),(3,1 ,-2),(2,0,-1)]`

Now , determinant of A is |A| = `[(1,2,3),(1 ,1, 5),(2 ,4 ,7)]`

= 1(7 - 20) - 2(7 - 10) + 3(4 - 2)

= 1(-13) - 2(-3) + 3(2)

= -13 + 6 + 6 = -1

∴ |A| = -1 ≠ 0

Now inverse of A is A^-1 = `1/|"A"|` × Adj A

= `1/-1 [(-13,-2,7),(3,1,-2),(2,0,-1)]`

∴ A^-1 = `[(13,2,-7),(-3,-1,2),(-2,0,1)]`