Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 1 Applications of Matrices and Determinants Ex 1.4 Text Book Back Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.4

Choose the most suitable answer from the given four alternatives:

Question 1.

A = (1, 2, 3), then the rank of AA^{T} is

(a) 0

(b) 2

(c) 3

(d) 1

Solution:

(d) 1

Hint:

A = (1, 2, 3) then

A^{T} = \(\left(\begin{array}{l}

1 \\

2 \\

3

\end{array}\right)\)

AA^{T} = (1, 2, 3) \(\left(\begin{array}{l}

1 \\

2 \\

3

\end{array}\right)\) = (1 + 4 + 9) = (14)

Number of non-zero rows = 1

∴ P (AA^{T}) = 1

Question 2.

The rank of m × n matrix whose elements are unity is

(a) 0

(b) 1

(c) m

(d) n

Solution:

(b) 1

Question 3.

If T = is a transition probability matrix, then at equilibriuium A is equal to

(a) \(\frac { 1 }{ 4 }\)

(b) \(\frac { 1 }{ 5 }\)

(c) \(\frac { 1 }{ 6 }\)

(d) \(\frac { 1 }{ 8 }\)

Solution:

(a) \(\frac { 1 }{ 4 }\)

Hint:

At equilibrium (A B) \(\left(\begin{array}{ll}

0.4 & 0.6 \\

0.2 & 0.8

\end{array}\right)\)

0.4 A + 0.2 B = A

0.4 A + 0.2(1 – A) = A

0.4 A + 0.2 – 0.2 A = A

0.2 A + 0.2 = A

0.2 = A – 0.2 A

0.2 = 0.8 A

A = \(\frac { 0.2 }{ 0.8 }\) = \(\frac { 1 }{ 4 }\)

Question 4.

If A = \(\left(\begin{array}{ll}

2 & 0 \\

0 & 8

\end{array}\right)\), then p(A) is

(a) 0

(b) 1

(c) 2

(d) n

Solution:

(c) 2

Hint:

A = \(\left(\begin{array}{ll}

2 & 0 \\

0 & 8

\end{array}\right)\)

No. of Non-zero rows = 2

∴ p(A) = 2

Question 5.

The rank of the matrix \(\left[\begin{array}{lll}

1 & 1 & 1 \\

1 & 2 & 3 \\

1 & 4 & 9

\end{array}\right]\) is

(a) 0

(b) 1

(c) 2

(d) 3

Solution:

(d) 3

Hint:

No. of Non-zero rows = 3

∴ p(A) = 3

Question 6.

The rank of the unit matrix of order n is

(a) n – 1

(b) n

(c) n + 1

(d) n²

Solution:

(b) n

Question 7.

If p(A) = r then which of the following is correct?

(a) all the minors of order r which does not vanish

(b) A has at least one minor of order r which does not vanish

(c) A has at least one (r + 1) order minor which vanishes

(d) all (r + 1) and higher order minors should not vanish

Solution:

(b) A has at least one minor of order r which does not vanish.

Question 8.

If A = \(\left(\begin{array}{l}

1 \\

2 \\

3

\end{array}\right)\) then the rank of AA^{T} is

(a) 0

(b) 1

(c) 2

(d) 3

Solution:

(b) 1

Hint:

No. of non-zero rows = 1

p (AA^{T}) = 1

Question 9.

If the rank of the matrix \(\left[\begin{array}{ccc}

\lambda & -1 & 0 \\

0 & \lambda & -1 \\

-1 & 0 & \lambda

\end{array}\right]\) is 2. then λ is

(a) 1

(b) 2

(c) 3

(d) only real number

Solution:

(a) 1

Hint:

A = \(\left[\begin{array}{ccc}

\lambda & -1 & 0 \\

0 & \lambda & -1 \\

-1 & 0 & \lambda

\end{array}\right]\)

Since the Rank is 2, third order matrix vanishes

∴ |A| = 0

\(\left[\begin{array}{ccc}

\lambda & -1 & 0 \\

0 & \lambda & -1 \\

-1 & 0 & \lambda

\end{array}\right]\) = 0

λ(λ² – 0) + 1 (0 – 1) = 0

λ³ – 1 = 0 ⇒ λ³ = 1

∴ λ = 1

Question 10.

(a) 0

(b) 2

(c) 3

(d) 5

Solution:

(c) 3

Hint:

Number of non-zero rows = 3

∴ Rank = 3

Question 11.

If is a transition probability matrix, then the value of x is

(a) 0.2

(b) 0.3

(c) 0.4

(d) 0.7

Solution:

(c) 0.4

Hint:

is a transition probability matrix then 0.6 + x = 1

x = 1 – 0.6

x = 0.4

Question 12.

Which of the following is not an elementary transformation?

(a) R_{i} ↔ Rj

(b) R_{i} → 2R_{i} + 2Cj

(c) R_{i} → 2R_{i} + 4Rj

(d) C_{i} → C_{i} + 5Cj

Solution:

(b) R_{i} → 2R_{i} + 2Cj

Question 13.

If p(A) = p(A,B)= then the system is

(a) Consistent and has infinitely many solutions

(b) Consistent and has a unique solution

(c) inconsistent

(d) consistent

Solution:

(d) Consistent

Question 14.

If p(A) = p(A,B)= the number of unknowns, then the system is

(a) Consistent and has infinitely many solutions

(b) Consistent and has a unique solution

(c) inconsistent

(d) consistent

Solution:

(b) Consistent and has a unique solution

Question 15.

If p(A) ≠ p(A, B) =, then the system is

(a) Consistent and has infinitely many solutions

(b) Consistent and has a unique solution

(c) inconsistent

(d) consistent

Solution:

(c) inconsistent

Question 16.

In a transition probability matrix, all the entries are greater than or equal to

(a) 2

(b) 1

(c) 0

(d) 3

Solution:

(c) 0

Question 17.

If the number of variables in a non-homogeneous system AX = B is n, then the system possesses a unique solution only when

(a) p(A) = p(A, B) > n

(b) p(A) = p(A, B) = n

(c) p(A) = p(A, B) < n

(d) none of these

Solution:

(b) p(A) = p(A, B) = n

Question 18.

The system of equations 4x + 6y = 5, 6x + 9y = 7 has

(a) a unique solution

(b) no solution

(c) infinitely many solutions

(d) none of these

Solution:

(b) no solution

Hint:

The matrix form of the equations

p(A) ≠ p(A, B)

∴ The system is inconsistent.

Question 19.

For the system of equations x + 2y + 3z = 1, 2x + y – z = 3, 3x + 2y + k = 4

(a) there is only one solution

(b) there exists infinitely many solution

(c) there is no solution

(d) none of these

Solution:

(a) there is only one solution

Hint:

The matrix form of the equations

The last equivalent matrix in the echelon form

p(A) = p (A, B) = no. of unknowns

∴ The system is consistent.

Question 20.

If |A| ≠ 0, then A is

(a) non – singular matrix

(b) singular matrix

(c) zero matrix

(d) none of these

Solution:

(a) non-singular matrix

Question 21.

The system of linear equations x = y + z = 2, 2x + y – z = 3, 3x + 2y + k = 4 has unique solution, if k is not equal to

(a) 1

(b) 0

(c) 3

(d) 7

Solution:

(b) 0

Hint:

The matrix form of the equations

Since the system has unique solution.

P(A) = p(A, B) ≠ n

∴ K ≠ 0

Question 22.

Cramer’s rule is applicable only to get an unique solution when

(a) Δ_{z} ≠ 0

(b) Δ_{x} ≠ 0

(c) Δ ≠ 0

(d) Δ_{y} ≠ 0

Solution:

(c) Δ ≠ 0

Question 23.

Then (x, y) is

Solution:

(d) (\(\frac { -Δ_1 }{ Δ_2 }\), \(\frac { -Δ_1 }{ Δ_3 }\))

Question 24.

|A_{n × n}| = 3 |adj A| = 243 = (3)5 then the value n is

(a) 4

(b) 5

(c) 6

(d) 7

Solution:

(c) 6

Question 25.

Rank of a null matrix is

(a) 0

(b) -1

(c) ∞

(d) 1

Solution:

(a) 0