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

Choose the correct answer.

Question 1.

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

(a) 0

(b) 2

(c) 3

(d) 1

Answer:

(d) 1

Hint:

Question 2.

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

(a) 0

(b) 1

(c) m

(d) n

Answer:

(b) 1

Hint:

All the rows except the first row can be made zero

Question 3.

If is a transition probability matrix, then at equilibrium A is equal to

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

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

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

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

Answer:

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

Hint:

Question 4.

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

2 & 0 \\

0 & 8

\end{array}\right)\) then ρ(A) is _______

(a) 0

(b) 1

(c) 2

(d) n

Answer:

(c) 2

Hint:

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

Answer:

(d) 3

Hint:

Question 6.

The rank of the unit matrix of order n is _______

(a) n – 1

(b) n

(c) n + 1

(d) n^{2}

Answer:

(b) n

Hint:

Unit matrix of order n is in echelon form with n non-zero rows

Question 7.

If ρ(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

Answer:

(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

Answer:

(b) 1

Hint:

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

Answer:

(a) 1

Hint:

Since rank is 2, the third order minor should vanish.

λ^{3} – 1 = 0

⇒ λ = 1

Question 10.

The rank of the diagonal matrix

is _______

(a) 0

(b) 2

(c) 3

(d) 5

Answer:

(c) 3

Hint:

There are only three non-zero rows as the matrix is in echelon form.

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

Answer:

(c) 0.4

Hint:

x = 1 – 0.6 = 0.4

Question 12.

Which of the following is not an elementary transformation?

(a) R_{i} ↔ R_{j}

(b) R_{i} → 2R_{i} + 2C_{j}

(c) R_{i} → 2R_{i} – 4R_{j}

(d) C_{i} → C_{i} + 5C_{j}

Answer:

(b) R_{i} → 2R_{i} + 2C_{j}

Hint:

Since rows and columns cannot be taken together.

Question 13.

If ρ(A) = ρ(A, B), then the system is _______

(a) Consistent and has infinitely many solutions

(b) Consistent and has unique solutions

(c) consistent

(d) inconsistent

Answer:

(c) consistent

Question 14.

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

(a) Consistent and has infinitely many solutions

(b) Consistent and has unique solutions

(c) inconsistent

(d) consistent

Answer:

(i) Consistent and has unique solutions

Question 15.

If ρ(A) ≠ ρ(A, B), then the system is ________

(a) Consistent and has infinitely many solutions

(b) Consistent and has unique solutions

(c) inconsistent

(d) consistent

Answer:

(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

Answer:

(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) ρ(A) = ρ(A, B) > n

(b) ρ(A) = ρ(A, B) = n

(c) ρ(A) = ρ(A, B) < n

(d) none of these

Answer:

(b) ρ(A) = ρ(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

Answer:

(b) no solution

Question 19.

For the system of equations x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4 _______

(a) there is only one solution

(b) there exists infinitely many solutions

(c) there is no solution

(d) none of these

Answer:

(a) there is only one solution

Hint:

By Cramer’s rule, there is only one solution

Question 20.

If |A| ≠ 0, then A is _______

(a) non- singular matrix

(b) singular matrix

(c) zero matrix

(d) none of these

Answer:

(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) 4

(b) 0

(c) -4

(d) 1

Answer:

(b) 0

Hint:

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

Answer:

(c) Δ ≠ 0

Question 23.

Answer:

Hint:

Question 24.

|A_{n×n}| = 3 |adj A| = 243 then the value n is _______

(a) 4

(b) 5

(c) 6

(d) 1

Answer:

(b) 5

Hint:

|adj A| = |A|^{n-1}, n is order of matrix

243 = 3^{n-1}

3^{4} = 3^{n-1}

n = 5

Question 25.

Rank of a null matrix is ______

(a) 0

(b) -1

(c) ∞

(d) 1

Answer:

(a) 0