2nd PUC Maths Question Bank Chapter 4 Determinants Ex 4.4

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Karnataka 2nd PUC Maths Question Bank Chapter 4 Determinants Ex 4.4

2nd PUC Maths Determinants NCERT Text Book Questions and Answers Ex 4.4

Write the minors and co-factors of the elements of following determinants

Question 1.
(i)
\(\left|\begin{array}{cc}{2} & {-4} \\{0} & {3}\end{array}\right|\)
Answer:

(ii)
\(\left|\begin{array}{ll}{a} & {c} \\{b} & {d}\end{array}\right|\)
Answer:

Question 2
(i)
\(\left|\begin{array}{lll}{\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} \\{\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} \\{\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}}\end{array}\right|\)
Answer:

(ii)
\(\left|\begin{array}{ccc}{1} & {0} & {4} \\{3} & {5} & {-1} \\{0} & {1} & {2}\end{array}\right|\)
Answer:

Question 3.
Using cofactors of elements of second row, Evaluate
\(\Delta\left|\begin{array}{lll}{1} & {0} & {0} \\{0} & {1} & {0} \\{0} & {0} & {1}\end{array}\right|\)
Answer:
C₂₁ = 0   C₂₂ = 1      C₂₃ = 0
∴ Δ A = 0 x 0 + 1 x 1 + 0 x 0 = 1

Question 4.
Using cofactors of elements of 3rd column
\(\Delta=\left|\begin{array}{lll}{\mathbf{1}} & {\mathbf{x}} & {\mathbf{y} \mathbf{z}} \\{\mathbf{1}} & {\mathbf{y}} & {\mathbf{z} \mathbf{x}} \\{\mathbf{1}} & {\mathbf{z}} & {\mathbf{x} \mathbf{y}}\end{array}\right|\)
Answer:
M₁₃ = (z – y) M₂₃ = (z-x) M₃₃ = (y-x)
C₁₃ = (z-y) C₂₃ = -(z-n), C₃₃ = (y – x)
Δ = yz (z – y) + zx{-(z – x)} + xy (y – x)
= yz² – y²z – z²x + zx² + xy² – x²y
x² (z – y) + z (y² – z²) + yz (z – y)
x² (z – y) + x (y+ z) (y- z) + yz (z – y)
(z – y) {x² – x (y + z) + yz}
(z – y) {x² – xy – xz + yz}
(z – y) {x (x – y) – z (x – y)}
(z – y) (x – y) (x – z) = (z – y) (y – x) (z – x)

Question 5.
\(\text { If } \Delta=\left|\begin{array}{lll}{\mathbf{a}_{11}} & {\mathbf{a}_{12}} & {\mathbf{a}_{13}} \\{\mathbf{a}_{21}} & {\mathbf{a}_{22}} & {\mathbf{a}_{23}} \\{\mathbf{a}_{31}} & {\mathbf{a}_{32}} & {\mathbf{a}_{33}}\end{array}\right|\) is a cofactors of a_ij then the calue of Δ is given by of
(A) a₁₁ A₃₁ + a₁₂ A₃₂ + a₁₃ A₃₃
(B) a_n A_n + a₁₂ A₂₁ + a₁₃ A₃₁
(C) a₂₁ A_u + a₂₂ A₁₂ + a₂₃ A₁₃
(D) a_n A_n + a₂₁ A₂₁ + a₃₁ A₃₁
Answer:
Answer is (D)
i.e., a₁₁ A₁₁ + a₂₁+ a₃₁ A₃₁