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

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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) n²
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.
λ³ – 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
3⁴ = 3^n-1
n = 5

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