Question 1: Write Minors and Cofactors of the elements of following determinants:

Answer

(i) The given determinant is .

Minor of element a_ij is M_ij.

∴ M₁₁ = minor of element a₁₁ = 3

M₁₂ = minor of element a₁₂ = 0

M₂₁ = minor of element a₂₁ = −4

M₂₂ = minor of element a₂₂ = 2

Cofactor of a_ij is A_ij = (−1)^{i + j} M_ij.

 

(ii) The given determinant is .

Minor of element a_ij is M_ij.

M₁₁ = minor of element a₁₁ = d

M₁₂ = minor of element a₁₂ = b

M₂₁ = minor of element a₂₁ = c

M₂₂ = minor of element a₂₂ = a

Cofactor of a_ij is A_ij = (−1)^{i + j} M_ij.

 

A₁₁ = (−1)¹⁺¹ M₁₁ = (−1)² (d) = d

A₁₂ = (−1)¹⁺² M₁₂ = (−1)³ (b) = −b

A₂₁ = (−1)²⁺¹ M₂₁ = (−1)³ (c) = −c

A₂₂ = (−1)²⁺² M₂₂ = (−1)⁴ (a) = a

 

Question 2:

Answer

(i) The given determinant is .

By the definition of minors and cofactors, we have:

M₁₁ = minor of a₁₁=

M₁₂ = minor of a₁₂=

M₁₃ = minor of a₁₃ =

M₂₁ = minor of a₂₁ =

M₂₂ = minor of a₂₂ =

M₂₃ = minor of a₂₃ =

M₃₁ = minor of a₃₁=

M₃₂ = minor of a₃₂ =

M₃₃ = minor of a₃₃ =

A₁₁ = cofactor of a₁₁= (−1)¹⁺¹ M₁₁ = 1

A₁₂ = cofactor of a₁₂ = (−1)¹⁺² M₁₂ = 0

A₁₃ = cofactor of a₁₃ = (−1)¹⁺³ M₁₃ = 0

A₂₁ = cofactor of a₂₁ = (−1)²⁺¹ M₂₁ = 0

A₂₂ = cofactor of a₂₂ = (−1)²⁺² M₂₂ = 1

A₂₃ = cofactor of a₂₃ = (−1)²⁺³ M₂₃ = 0

A₃₁ = cofactor of a₃₁ = (−1)³⁺¹ M₃₁ = 0

A₃₂ = cofactor of a₃₂ = (−1)³⁺² M₃₂ = 0

A₃₃ = cofactor of a₃₃ = (−1)³⁺³ M₃₃ = 1

 

(ii) The given determinant is .

By definition of minors and cofactors, we have:

M₁₁ = minor of a₁₁=

M₁₂ = minor of a₁₂=

M₁₃ = minor of a₁₃ =

M₂₁ = minor of a₂₁ =

M₂₂ = minor of a₂₂ =

M₂₃ = minor of a₂₃ =

M₃₁ = minor of a₃₁=

M₃₂ = minor of a₃₂ =

M₃₃ = minor of a₃₃ =

A₁₁ = cofactor of a₁₁= (−1)¹⁺¹ M₁₁ = 11

A₁₂ = cofactor of a₁₂ = (−1)¹⁺² M₁₂ = −6

A₁₃ = cofactor of a₁₃ = (−1)¹⁺³ M₁₃ = 3

A₂₁ = cofactor of a₂₁ = (−1)²⁺¹ M₂₁ = 4

A₂₂ = cofactor of a₂₂ = (−1)²⁺² M₂₂ = 2

A₂₃ = cofactor of a₂₃ = (−1)²⁺³ M₂₃ = −1

A₃₁ = cofactor of a₃₁ = (−1)³⁺¹ M₃₁ = −20

A₃₂ = cofactor of a₃₂ = (−1)³⁺² M₃₂ = 13

A₃₃ = cofactor of a₃₃ = (−1)³⁺³ M₃₃ = 5

 

Question 3:

Using Cofactors of elements of second row, evaluate .

Answer

The given determinant is .

We have:

∴ A₂₁ = cofactor of a₂₁ = (−1)²⁺¹ M₂₁ = 7

∴ A₂₂ = cofactor of a₂₂ = (−1)²⁺² M₂₂ = 7

∴ A₂₃ = cofactor of a₂₃ = (−1)²⁺³ M₂₃ = −7

We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.

∴ Δ = a₂₁A₂₁ + a₂₂A₂₂ + a₂₃A₂₃ = 2(7) + 0(7) + 1(−7) = 14 − 7 = 7

 

Question 4:

Using Cofactors of elements of third column, evaluate

Answer

The given determinant is .

We have:

∴ A₁₃ = cofactor of a₁₃ = (−1)¹⁺³ M₁₃ = (z − y)

A₂₃ = cofactor of a₂₃ = (−1)²⁺³ M₂₃ = − (z − x) = (x − z)

A₃₃ = cofactor of a₃₃ = (−1)³⁺³ M₃₃ = (y − x)

We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.

∴ Δ = a₁₃A₁₃ + a₂₃A₂₃ + a₃₃A₃₃

= yz(z - y) + zx(x - z) + xy(y - x)

= yz² – y²z + x²z – xz² + xy² – x²y

= (x²z – y²z) + (yz² – xz²) + (xy² – x²y)

= z (x² – y²) + z²  (y – x) + xy (y – x)

= (x - y) [zx + zy – z² - xy]

= (x - y) [z (x - z) + y (z - x)]

= (x - y) (z - x)[-z + y]

= (x - y) (y – z) (z - x)

Hence, Δ = (x – y) (y - z) (z - x).

 

Question 5: For the matrices A and B, verify that (AB)′ = B’A’ where

Answer

Hence, we have verified that (AB)’ = B’A’.

Hence, we have verified that (AB)’ = B’A’.