Answer :

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 A_{ij} denote the cofactors of Matrix A

Minor of an element a_{ij} = M_{ij �}

a_{11} = 1, Minor of element a_{11} = M_{11} = = (3 × 5) – (1 × 1) = 14

a_{12} = -2, Minor of element a_{12} = M_{12} = = (-2 × 5) – (1 × 1) = -11

a_{13} = 1, Minor of element a_{13} = M_{13} = = (-2 × 1) – (3 × 1) = -5

a_{21} = -2, Minor of element a_{21} = M_{21} = = ((-2) × 5) – (1 × 1) = -11

a_{22} = 3, Minor of element a_{22} = M_{22} = = (1 × 5) – (1 × 1) = 4

a_{23} = 1, Minor of element a_{23} = M_{23} = = (1 × 1) – ((-2) × 1) = 3

a_{31} = 1, Minor of element a_{31} = M_{31} = = (-2 × 1) – (3 × 1) = -5

a_{32} = 1, Minor of element a_{32} = M_{32} = = (1 × 1) – (1 × (-2)) = 3

a_{33} = 5, Minor of element a_{33} = M_{33} = = (1 × 3) – ((-2) × (-2)) = -1

Cofactor of an element a_{ij} = A_{ij}

A_{11} = (-1)^{1+1}× 14 = 1 × 14 = 14

A_{12} = (-1)^{1+2}× (-11) = (-1) × (-11) = 11

A_{13} = (-1)^{1+3}× (-5) = 1 × (-5) = -5

A_{21} = (-1)^{2+1}× (-11) = (-1) × (-11) = 11

A_{22} = (-1)^{2+2} × 4 = 1 × 4 = 4

A_{23} = (-1)^{2+3} × 3 = (-1) × 3 = -3

A_{31} = (-1)^{3+1} × (-5) = 1 × (-5) = -5

A_{32} = (-1)^{3+2} × 3 = (-1) × 3 = -3

A_{33} = (-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 a_{ij} = A_{ij}

A_{11} = (-1)^{1+1}× (-1/13) = 1 × (-1/13) = -1/13

A_{12} = (-1)^{1+2}× (-2/13) = (-1) × (-2/13) = 2/13

A_{13} = (-1)^{1+3}× (-1/13) = 1 × (-1/13) = -1/13

A_{21} = (-1)^{2+1}× (-2/13) = (-1) × (-2/13) = 2/13

A_{22} = (-1)^{2+2} × (3/13) = 1 × (-3/13) = -3/13

A_{23} = (-1)^{2+3} × (1/13) = (-1) × (1/13) = -1/13

A_{31} = (-1)^{3+1} × (-1/13) = 1 × (-1/13) = -1/13

A_{32} = (-1)^{3+2} × (1/13) = (-1) × 1/13 = -1/13

A_{33} = (-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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