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𝐒𝐂𝐎𝐑𝐄 − 𝐈𝐈 𝐄𝐍𝐓𝐇𝐔𝐒𝐈𝐀𝐒𝐓 𝐂𝐎𝐔𝐑𝐒𝐄

GR-19 : MATRICES AND DETERMINANTS

SINGLE CORRECT TYPE QUESTIONS

Q.(1): Let 𝑀 denotes the set of all 2 × 2 matrices. Define the relation

𝑅 = L(𝐴, 𝐵) ∈ 𝑀 × 𝑀 Q 𝐴 + 𝐵! is a symmetric matrixU .

Then R is
(A) symmetric, transitive but not reflexive relation
(B) reflexive, symmetric but not transitive relation
(C) reflexive, transitive but not symmetric relation
(D) an equivalence relation

Q.(2): If 𝐴, 𝐵, 𝐶 are three matrices of order 2, such that

|𝐴 + 𝐵| − |𝐶| = −10,
|𝐵 + 𝐶| − |𝐴| = 25,
|𝐶 + 𝐴| − |𝐵| = 15,
then the value of |𝐴 + 𝐵 + 𝐶| is equal to (|A| denotes the determinant of matrix A )
(A) 15 (B) 30 (C) 45 (D) 60

Q.(3): If |A| denotes the determinant of matrix A, where A is a square matrix of order 3, satisfying
𝑡𝑟(𝐴) = 4, 𝑡𝑟(𝐴" ) = 10, 𝑎𝑛𝑑 𝑡𝑟(𝐴# ) = 28,
then find the value of |𝐴|.
(A) 1 (B) 2 (C) 3 (D) 0

MULTIPLE CORRECT TYPE QUESTIONS

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𝐒𝐂𝐎𝐑𝐄 − 𝐈𝐈 𝐄𝐍𝐓𝐇𝐔𝐒𝐈𝐀𝐒𝐓 𝐂𝐎𝐔𝐑𝐒𝐄

Q.(4):

Q.(5):

Q.(6): Which of the following option(s) represents a possible value k, satisfying

"$"%
1 1 1 1 (
⎡1 + 1− ⎤ &$&#'#( ⎡ 1− 1+ ⎤
√3 √3⎥ 2 "
√3 √3
⎢ = y! z ⎢ ⎥ .
⎢ 1 1 √3 ⎢−1 − 1 −1 + 1 ⎥
−1 1+ ⎥
⎣ √3 √3⎦ ⎣ √3 √3⎦
(A) 12173 (B) 48413 (C) 12217 (D) 24265

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𝐒𝐂𝐎𝐑𝐄 − 𝐈𝐈 𝐄𝐍𝐓𝐇𝐔𝐒𝐈𝐀𝐒𝐓 𝐂𝐎𝐔𝐑𝐒𝐄

Q.(7): Given that 𝑎, 𝑏, 𝑐 are non-zero real and distinct constants, such that
𝑥 " − 𝑎" 𝑥 " − 𝑏" 𝑥" − 𝑐"
•(𝑥 − 𝑎)# (𝑥 − 𝑏)# (𝑥 − 𝑐)# • = 0.
(𝑥 + 𝑎)# (𝑥 + 𝑏)# (𝑥 + 𝑐 ) #
If 𝑎 = 11, 𝑏 = 12, 𝑐 = 13, then the given equation has m solutions.
If 𝑎 = 10, 𝑏 = −12, 𝑐 = 3, then the given equation has n solutions.
If 𝑎 = 60, 𝑏 = −12, 𝑐 = 15 then the given equation has p solutions.
Which of the following options is/are correct?
(A) 2𝑚 + 3𝑛 = 9 (B) 3𝑛 + 4𝑝 = 7 (C) 5𝑚 − 6𝑝 = 4 (D)3𝑛 + 4𝑝 = 4

Q.(8): Let 𝑎, 𝑏, 𝑐 ∈ ℝ, 𝑏 ≠ 𝑐 ≠ 0, such that

𝑏𝑐 − 4𝑎" 2𝑎𝑐 − 𝑏 " 2𝑎𝑏 − 𝑐 "
Š2𝑎𝑐 − 𝑏 " 2𝑎𝑏 − 𝑐 " 𝑏𝑐 − 4𝑎" Š = 0.
2𝑎𝑏 − 𝑐 " 𝑏𝑐 − 4𝑎" 2𝑎𝑐 − 𝑏 "
Then, which of the following options is/are correct for all possible values of a, b, c?
%
(A) 24𝑎𝑥 " + 4𝑏𝑥 + 𝑐 = 0 has at least one root in ‹0, " Œ

(B) 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 always passes through a fixed point

(C) 𝑎𝑥 " + 𝑏𝑥 + 𝑐 = 0 has at least one root in (0, 2)
(D) 6𝑎𝑥 " + 2𝑏𝑥 + 𝑐 = 0 has at least one root in (0, 1)

Q.(9):
Let A and B are any two square matrices of order 2 × 2 , with real entries, such that
𝐴𝐵 = 𝐴" 𝐵" − (𝐴𝐵)" 𝑎𝑛𝑑 |𝐵| = 3.
If |A| denotes the determinant of matrix A, then which of the following options is/are correct?
(A) |𝐴 + 𝐼| = 1 (B) |𝐴 + 3𝐵| − |3𝐴 + 𝐵| = 24
(C) |2𝐴 + 𝐵| − |𝐴 + 2𝐵| = 9 (D) 𝑎𝑑𝑗(𝐴𝐵) = 𝑂.

Q.(10): Let A be a square matrix of order 3, satisfying 𝐴%) = 𝐴 + 𝐼, then which of the following
options is/are correct(|A| denotes the determinant of matrix A )?
(A) |𝐴#* − 𝐼| > 0 (B) |𝐴 − 𝐼| < 0 (C) |𝐴 + 𝐼| < 0 (D) |𝐴| > 0

Q.(11): For 3 × 3 matrices M and N, which of the following statement(s) is(are) NOT correct ?
(A) 𝑁 ! 𝑀𝑁 is symmetric or skew symmetric, according as 𝑀 is symmetric or skew symmetric.
(B) 𝑀𝑁 − 𝑁𝑀 is skew symmetric for all symmetric matrices 𝑀 and 𝑁.

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𝐒𝐂𝐎𝐑𝐄 − 𝐈𝐈 𝐄𝐍𝐓𝐇𝐔𝐒𝐈𝐀𝐒𝐓 𝐂𝐎𝐔𝐑𝐒𝐄

(C) 𝑀𝑁 is symmetric for all symmetric matrices 𝑀 and 𝑁.

(D) adj 𝑀 adj 𝑁 = adj 𝑀𝑁 for all square matrices 𝑀 and 𝑁of same order.

NUMERICAL ANSWER TYPE QUESTIONS

Q.(12): The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 × 2
matrix such that the trace of A is 3 and the trace of 𝐴3 is −18, then the value of the determinant of
A is

+
Q.(13): Let 𝑀 = ™ sin 𝑥" −1 − sin" 𝑥š = 𝑓(𝑥)𝐼 + 𝑔(𝑥)𝑀'% , where 𝐼 is an identity matrix of
" "
1 + cos 𝑥 cos + 𝑥
order 2,
𝑓, = min{𝑓(𝑥)|𝑥 ∈ [0,2𝜋)} , and 𝑓- = max{𝑓(𝑥)|𝑥 ∈ [0,2𝜋)}.
Then the value of 1 − 16(𝑓, + 𝑓- ) is equal to

Q.(14): How many 3 × 3 matrices M with entries from {0,1,2} are there, for which the sum of the
diagonal entries of MTM is 5 ?

Q.(15): For a real number ‘a’, if the system

1 𝑎 𝑎" 𝑥 1
£𝑎 1 𝑎 ¤ ¥𝑦§ = £−1¤
𝑎" 𝑎 1 𝑧 1
of linear equations, has infinitely many solutions, then 1 + a + a2 equals

Q.(16):

Q.(17): If 𝑎, 𝑏, 𝑐, 𝑥. , 𝑦. , 𝑧. are real numbers (𝑖 = 1,2,3) satisfying

𝑎𝑥%" + 𝑏𝑦%" + 𝑐𝑧%" = 36, 𝑎𝑥" 𝑥# + 𝑏𝑦" 𝑦# + 𝑐𝑧" 𝑧# = 21,
" " "
©𝑎𝑥" + 𝑏𝑦" + 𝑐𝑧" = 25,ª , 𝑎𝑏𝑐 = 8, « 𝑎𝑥% 𝑥# + 𝑏𝑦% 𝑦# + 𝑐𝑧% 𝑧# = 20, ¬ ,
𝑎𝑥#" + 𝑏𝑦#" + 𝑐𝑧#" = 20, 𝑎𝑥" 𝑥% + 𝑏𝑦" 𝑦% + 𝑐𝑧" 𝑧% = 23

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𝐒𝐂𝐎𝐑𝐄 − 𝐈𝐈 𝐄𝐍𝐓𝐇𝐔𝐒𝐈𝐀𝐒𝐓 𝐂𝐎𝐔𝐑𝐒𝐄

𝑦" 𝑧# − 𝑦# 𝑧" 𝑥# 𝑧" − 𝑥" 𝑧# 𝑦# 𝑥" − 𝑦" 𝑥#

then find the value of Š𝑦# 𝑧% − 𝑦% 𝑧# 𝑥% 𝑧# − 𝑥# 𝑧% 𝑦% 𝑥# − 𝑦# 𝑥% Š.
𝑦% 𝑧" − 𝑦" 𝑧% 𝑥" 𝑧% − 𝑥% 𝑧" 𝑦" 𝑥% − 𝑦% 𝑥"

Q.(18): Let 𝐴 = -𝑎./ ® , and 𝑎𝑑𝑗𝐴 = -𝑐./ ®#×# be two matrices satisfying
#×#

𝐴𝐴! = 4𝐼# , 2𝑎./ + 𝑐./ = 0∀𝑖, 𝑗 𝑎𝑛𝑑 𝜆 = 3|𝐴 + 𝐼# | + |𝐴 + 2𝐼# | + |𝐴 + 3𝐼# |.

Then the value of |𝜆| is equal to

PASSAGES

Passage-I:
Let
# #

𝑋 = ³-𝑎./ ®#×# : 𝑎./ = 𝑎/. , 𝑎./ ∈ {0,1}, ´ ´ 𝑎./ = 5µ ,

.1% /1%

𝑥 1
𝑌 = ·-𝑏./ ®#×# : 𝐵 = -𝑏./ ®#×# , 𝐵 ∈ 𝑋, 𝐵 ¥𝑦§ = £0¤ has unique solution¸ ,
𝑧 0
𝑥 1
𝑍 = ·-𝑐./ ®#×# : 𝐶 = -𝑐./ ®#×# , 𝐶 ∈ 𝑋, 𝐶 ¥𝑦§ = £0¤ has NO solution¸ ,
𝑧 0
𝑥 1
𝑇 = ·-𝑑./ ®#×# : 𝐷 = -𝑑./ ®#×# , 𝐷 ∈ 𝑋, 𝐷 ¥𝑦§ = £0¤ has infinite solutions¸ .
𝑧 0
Now, answer the following questions:

Q.(19): Which of the following options is/are correct?

(A) 𝑛(𝑋) = 12 (B) 𝑛(𝑌) = 6 (C) 𝑛(𝑍) = 5 (D) 𝑛(𝑇) = 3

Q.(20): Which of the following options is/are correct?

(A) 𝑛(𝑋) = 13 (B) 𝑛(𝑌) = 7 (C) 𝑛(𝑍) = 4 (D) 𝑛(𝑇) = 2

Passage-II:
If A and B are two square matrices of order n, then answer the following questions:

Q.(21): If A, B and A+B are idempotent matrices, then

(A) there exists A, B such that 𝐴𝐵 ≠ 𝑂. (B) 𝐵𝐴 = 𝑂
(C) there exists A, B such that 𝐵𝐴 ≠ 𝑂. (D) 𝐴𝐵 = 𝑂

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𝐒𝐂𝐎𝐑𝐄 − 𝐈𝐈 𝐄𝐍𝐓𝐇𝐔𝐒𝐈𝐀𝐒𝐓 𝐂𝐎𝐔𝐑𝐒𝐄

Q.(22): Which of the following options is/are correct?

(A) If A and B are orthogonal matrices, then BA is also orthogonal.
(B) If A and B are orthogonal matrices, then AB is also orthogonal.
(C) If 𝐴 + 𝐵 = 𝐼, and A is an idempotent matrix, then 𝐴𝐵 = 𝐵𝐴 = 𝑂.
(D) If 𝐴" + 𝐵" = 𝐴𝐵, then 𝑡𝑟(𝐴𝐵𝐴𝐵) = 𝑡𝑟(𝐴" 𝐵" ).

MATRIX MATCHING

Q.(23): If matrices A and B are defined as

𝑛, 𝑖=𝑗 −𝑏𝑐 𝑏 " + 𝑏𝑐 𝑏 " + 𝑏𝑐

𝐴2,, = -𝑎./ ®2×2 , 𝑎./ = ¼ 𝑎𝑛𝑑 𝐵 = £ 𝑐 " + 𝑐𝑎 −𝑐𝑎 𝑐 " + 𝑐𝑎 ¤ ,
𝑚 𝑖≠𝑗
𝑎" + 𝑎𝑏 𝑎" + 𝑎𝑏 −𝑎𝑏
where a, b, c are roots of 3𝑥 # + 11 𝑥 " − 21 𝑥 − 29 = 0. Then, match the following columns
Column-I Column-II
(A) gcd½¾det 𝐴&,# ¾, |det 𝐵|¿ (P) 1
(B) gcd½¾det 𝐴4,%4 ¾, |8 + det 𝐵|¿ (Q) 3
(C) gcd½¾det 𝐴%%,5 ¾, 2 − det 𝐵¿ (R) 5
(D) gcd½¾det 𝐴*,%% ¾, |2 + det 𝐵|¿ (S) 7

KEY

1. D 2. B 3. D 4. 234 5. 3,4
6. CD 7. AB 8. ABCD 9. AB 10. AD
11. CD 12. 7 13. 30 14. 198 15. 1
16. 3 17. 108 18. 2 19. AB 20. CD
21. BD 22. ABCD 23. A-S,B-R,C-Q,D-P 24. 25.

E-6 MATHEMATICS