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 A_ij denote the cofactors of Matrix A

Minor of an element a_ij = M_{ij �}

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 a_ij = A_ij

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 a_ij = A_ij

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