Let . Verify that

(A-1)-1 = A


A =

|A| = 1 (3 × 5 – 1 × 1) – (-2) ((-2) × 5 – 1 × 1) + 1 ((-2) × 1 – 3 × 1)


|A| = (15 – 1) + 2 (-10 – 1) + (-2 – 3)


|A| = 14 – 22 – 5 = -13


To find the inverse of a matrix we need to find the Adjoint of that matrix


For finding the adjoint of the matrix we need to find its cofactors


Let Aij denote the cofactors of Matrix A


Minor of an element aij = Mij �


a11 = 1, Minor of element a11 = M11 = = (3 × 5) – (1 × 1) = 14


a12 = -2, Minor of element a12 = M12 = = (-2 × 5) – (1 × 1) = -11


a13 = 1, Minor of element a13 = M13 = = (-2 × 1) – (3 × 1) = -5


a21 = -2, Minor of element a21 = M21 = = ((-2) × 5) – (1 × 1) = -11


a22 = 3, Minor of element a22 = M22 = = (1 × 5) – (1 × 1) = 4


a23 = 1, Minor of element a23 = M23 = = (1 × 1) – ((-2) × 1) = 3


a31 = 1, Minor of element a31 = M31 = = (-2 × 1) – (3 × 1) = -5


a32 = 1, Minor of element a32 = M32 = = (1 × 1) – (1 × (-2)) = 3


a33 = 5, Minor of element a33 = M33 = = (1 × 3) – ((-2) × (-2)) = -1


Cofactor of an element aij = Aij


A11 = (-1)1+1× 14 = 1 × 14 = 14


A12 = (-1)1+2× (-11) = (-1) × (-11) = 11


A13 = (-1)1+3× (-5) = 1 × (-5) = -5


A21 = (-1)2+1× (-11) = (-1) × (-11) = 11


A22 = (-1)2+2 × 4 = 1 × 4 = 4


A23 = (-1)2+3 × 3 = (-1) × 3 = -3


A31 = (-1)3+1 × (-5) = 1 × (-5) = -5


A32 = (-1)3+2 × 3 = (-1) × 3 = -3


A33 = (-1)3+3 × (-1) = 1 × (-1) = -1


Adj A = =


A-1 = (Adj A)/|A|


A-1 = =


(ii) To find (A-1)-1 we have to find out Adj(A-1)


A-1 =


|A-1| = (-1/13)3 [14 (4 × (-1) – (-3) × (-3)) – 11 (11 × (-1) – (-3) × (-5)) + (-5) (11 × (-3) – 4 × (-5))]


|A| = (-1/13)3 [14 (-4 – 9) – 11 (-11 – 15) – 5 (-33 + 20)]


|A| = (-1/13)3 [14 × (-13) – 11 × (-26) – 5 × (-13)]


|A| = (-1/13)3 × 169 = -1/13


Cofactor of an element aij = Aij


A11 = (-1)1+1× (-1/13) = 1 × (-1/13) = -1/13


A12 = (-1)1+2× (-2/13) = (-1) × (-2/13) = 2/13


A13 = (-1)1+3× (-1/13) = 1 × (-1/13) = -1/13


A21 = (-1)2+1× (-2/13) = (-1) × (-2/13) = 2/13


A22 = (-1)2+2 × (3/13) = 1 × (-3/13) = -3/13


A23 = (-1)2+3 × (1/13) = (-1) × (1/13) = -1/13


A31 = (-1)3+1 × (-1/13) = 1 × (-1/13) = -1/13


A32 = (-1)3+2 × (1/13) = (-1) × 1/13 = -1/13


A33 = (-1)3+3 × (-5/13) = 1 × (-5/13) = -5/13


Adj (A-1) = =


(A-1)-1 = Adj(A-1)/|A-1|


(A-1)-1 = = = A


(A-1)-1 = A


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