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

Question 1.
If A = (1 2 3), then the rank of AAT is ______
(a) 0
(b) 2
(c) 3
(d) 1
(d) 1
Hint:

Question 2.
The rank of m × n matrix whose elements are unity is _________
(a) 0
(b) 1
(c) m
(d) n
(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) $$\frac{1}{8}$$
(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
(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
(d) 3
Hint:

Question 6.
The rank of the unit matrix of order n is _______
(a) n – 1
(b) n
(c) n + 1
(d) n2
(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
(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 AAT is _______
(a) 0
(b) 1
(c) 2
(d) 3
(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
(a) 1
Hint:

Since rank is 2, the third order minor should vanish.
λ3 – 1 = 0
⇒ λ = 1

Question 10.
The rank of the diagonal matrix
is _______
(a) 0
(b) 2
(c) 3
(d) 5
(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
(c) 0.4
Hint:
x = 1 – 0.6 = 0.4

Question 12.
Which of the following is not an elementary transformation?
(a) Ri ↔ Rj
(b) Ri → 2Ri + 2Cj
(c) Ri → 2Ri – 4Rj
(d) Ci → Ci + 5Cj
(b) Ri → 2Ri + 2Cj
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
(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
(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
(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
(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
(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
(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
(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
(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
(b) 0
Hint:

Question 22.
Cramer’s rule is applicable only to get an unique solution when ______
(a) Δz ≠ 0
(b) Δx ≠ 0
(c) Δ ≠ 0
(d) Δy ≠ 0
(c) Δ ≠ 0

Question 23.

Hint:

Question 24.
|An×n| = 3 |adj A| = 243 then the value n is _______
(a) 4
(b) 5
(c) 6
(d) 1
(b) 5
Hint:
|adj A| = |A|n-1, n is order of matrix
243 = 3n-1
34 = 3n-1
n = 5

Question 25.
Rank of a null matrix is ______
(a) 0
(b) -1
(c) ∞
(d) 1