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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.8

Choose the correct or the most suitable answer from the given four alternatives.

Question 1.

If |adj(adj A)| = |A|^{9}, then the order of the square matrix A is _______

(a) 3

(b) 4

(c) 2

(d) 5

Answer:

(b) 4

Question 2.

If A is a 3 × 3 non-singular matrix such that AA^{T} = A^{T}A and B = A^{-1}A^{T}, then BB^{T} = ______

(a) A

(b) B

(c) I_{3}

(d) B^{T}

Answer:

(c) I_{3}

Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

If A = \(\left[\begin{array}{ll}{2} & {0} \\ {1} & {5}\end{array}\right]\) and B = \(\left[\begin{array}{ll}{1} & {4} \\ {2} & {0}\end{array}\right]\) then |adj(AB)| = _______

(a) -40

(b) -80

(c) -60

(d) -20

Answer:

(b) -80

Hint:

Question 7.

If P = \(\left[\begin{array}{ccc}{1} & {x} & {0} \\ {1} & {3} & {0} \\ {2} & {4} & {-2}\end{array}\right]\) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is ______

(a) 15

(b) 12

(c) 14

(d) 11

Answer:

(d) 11

Hint: Given |A| = 4 and P is the adjoint matrix of A

|P| = 4^{2} = 16

⇒ -2 (3 – x) = 16

⇒ -6 + 2x = 16

⇒ 2x = 22

⇒ x = 11

Question 8.

Answer:

Question 9.

If A, B and C are invertible matrices of some order, then which one of the following is not true?

(a) adj A =|A| A^{-1}

(b) adj(AB) = (adj A) (adj B)

(c) det A^{-1} = (det A)^{-1}

(d) (ABC)^{-1} = C^{-1}B^{-1}A^{-1}

Answer:

(b) adj(AB) = (adj A) (adj B)

Question 10.

Answer:

Question 11.

If A^{T} A^{-1} is symmetric, then A^{2} = _______

(a) A^{-1}

(b) (A^{T})^{2}

(c) A^{T}

(d) (A^{-1})^{2}

Answer:

(b) (A^{T})^{2}

Question 12.

If A is a non-singular matrix such that \(A^{-1}=\left[\begin{array}{rr}{5} & {3} \\ {-2} & {-1}\end{array}\right]\), then (A^{T})^{-1} = _______

Answer:

Question 13.

Answer:

Question 14.

Answer:

Question 15.

Answer:

Question 16.

Answer:

Question 17.

Answer:

Question 18.

Answer:

Question 19.

Answer:

Question 20.

Which of the following is/are correct?

(i) Adjoint of a symmetric matrix is also a symmetric matrix

(ii) Adjoint of a diagonal matrix is also a diagonal matrix.

(iii) If A is a square matrix of order n and λ, is a scalar, then adj(λA) = λ^{n} adj(A).

(iv) A(adj A) = (adj A)A = |A| I

(a) Only (i)

(b) (ii) and (iii)

(c) (iii) and (iv)

(d) (i), (ii) and (iv)

Answer:

(d) (i), (ii) and (iv)

Question 21.

If ρ(A) = ρ([A|B]), then the system AX = B of linear equations is ______

(a) consistent and has a unique solution

(b) consistent

(c) consistent and has infinitely many solution

(d) inconsistent

Answer:

(b) consistent

Question 22.

If 0 ≤ θ ≤ π, the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ)x – y + z = 0, (sin θ)x + y – z = 0 has a non-trivial solution then θ is ______

(a) \(\frac{2 \pi}{3}\)

(b) \(\frac{3 \pi}{4}\)

(c) \(\frac{5 \pi}{4}\)

(d) \(\frac{\pi}{4}\)

Answer:

(d) \(\frac{\pi}{4}\)

Question 23.

The augmented matrix of a system of linear equations is \(\left[\begin{array}{cccc}{1} & {2} & {7} & {3} \\ {0} & {1} & {4} & {6} \\ {0} & {0} & {\lambda-7} & {\mu+5}\end{array}\right]\). The system has infinitely many solutions if _____

(a) λ = 7, μ ≠ 5

(b) λ = -7, μ = 5

(c) λ ≠ 7, μ ≠ -5

(d) λ = 7, μ = -5

Answer:

(d) λ = 7, μ = -5

Question 24.

Answer:

Question 25.

Answer:

### Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.8 Additional Problems

Question 1.

(a) m < n (b) m > n

(c) m = n

(d) None of these

Solution:

(c) m = n

Hint:

A = [a_{ij}]_{m×n} is a square matrix if number of rows is equal to that of columns, i.e., m = n.

Question 2.

Matrices A and B will be inverse of each other only if ……..

(a) AB = BA

(b) AB = BA = O

(c) AB = O, BA = I

(d) AB = BA = I

Solution:

(d) AB = BA = I

Hint:

By the definition of invertible matrices two matrices A and B are inverse of each other if AB = BA = I.

Question 3.

If A is an invertible matrix of order 2 then det (A^{-1}) is equal to

Solution:

Hint.

Question 4.

Solution:

(d) \(\frac{1}{-3}\left(\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right)\)

Question 5.

Solution:

(d) Inverse does not exist

Question 6.

Given ρ(A, B) = ρ(A) = number of unknowns, then the system has …….

(a) unique solution

(b) no solution

(c) inconsistent

(d) infinitely many solution

Solution:

(a) unique solution

Question 7.

Given ρ(A, B) ≠ ρ(A) then the system has

(a) no solution

(b) unique solution

(c) infinitely many solution

(d) None

Solution:

(a) no solution

Question 8.

Given ρ(A, B) = ρ(A) < number of unknowns, then the system has …….

(a) unique solution

(b) no solution

(c) 3 solutions

(d) infinitely many solution

Solution:

(d) infinitely many solution

Question 9.

(a) 1

(b) 2

(c) 3

(d) None of these

Solution:

(a) 1

Question 10.

(a) 1

(b) 0

(c) 4

(d) any real number

Solution:

(d) any real number

Question 11.

If A = [2 0 1], then the rank of AA^{T} is ………

(a) 1

(b) 2

(c) 3

(d) 0

Solution:

(a) 1

Hint.

Question 12.

If the rank of the matrix is 2, then λ is ……

(a) 1

(b) 2

(c) 3

(d) any real number

Solution:

(a) 1

Hint.

Since the rank of the given matrix is 2, the value of 3^{rd} order determinant is zero.

Question 13.

If A is a scalar matrix with scalar k ≠ 0, of order 3, then A^{-1} is …….

Solution:

\(\frac{1}{k} \mathrm{I}\)

Hint:

Question 14.

Solution:

Hint:

(adj A)(A) = |A|I

Question 15.

If A is a square matrix of order n, then |adj A| is …….

(a) |A|^{2}

(b) |A|^{n}

(c) |A|^{n – 1}

(d) |A|

Solution:

(c) |A|^{n – 1}

Hint:

When A is square matrix of order 3, then |adj A| = |A|^{2} = |A|^{3 – 1} ∴ When A is a square matrix of order n, |adj A| = |A|^{n – 1}

Question 16.

If A is a matrix of order 3, then det (kA)

(a) k^{3}(det A)

(b) k^{2}(det A)

(c) k(det A)

(d) det (A)

Solution:

(a) k^{3}(det A)

Hint:

A is a matrix of order 3. ∴ det (kA) = k^{3} (det A)

Question 17.

If I is the unit matrix of order n, where k ± 0 is a constant, then adj(kI) = …….

(a) k^{n}(adj I)

(b) k(adj I)

(c) k^{2}(adj I)

(d) k^{n – 1} (adj I)

Solution:

(d) k^{n – 1} (adj I)

Hint:

When I is a unit matrix of order 3, then adj (kI) = k^{2} (adj I)

∴ When I is a unit matrix of order n, then adj (kI) = k^{n – 1} (adj I)

Question 18.

In a system of 3 linear non-homogeneous equations with three unknowns, if ∆ = 0 and ∆x = 0, ∆y ≠ 0 and ∆z = 0, then the system has …..

(a) unique solution

(b) two solutions

(c) infinitely many solutions

(d) no solution

Solution:

(d) no solution

Hint:

When ∆ = 0 ∆x, ∆z = 0 but ∆y ≠ 0 ⇒ that the system is inconsistent.

∴ There is no solution.