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Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced


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12 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced

Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced for JEE 2022 is part of Maths 35 Years JEE Main & Advanced Past year Papers preparation. The Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced questions and answers have been prepared according to the JEE exam syllabus.The Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced MCQs are made for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced below.
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*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 1

The determinant  zero, if

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 1


Operating C3→ C3 – C1 α – C2, we get


⇒ (ac – b2) (aα2 + 2bα + c) = 0
⇒ either ac – b2 = 0 or aα2 + 2b α + c = 0
⇒ either a, b, c are in G.P. or (x – α) is a factor of ax2 + 2bx + c
⇒ (b) and (e) are the correct answers.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 2

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 2


[∴ C2 and C3 are identical]
⇒ x + iy = 0   ⇒ x = 0, y = 0

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 3

Let M and N be two 3 × 3 non-singular skew- symmetric matrices such that MN = NM. If  PT denotes the transpose of P, then M2N2 (MTN)–1 (MN–1)T is equal to

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 3

[As a skew symmetric matrix of order 3 cannot be non singular, therefore the data given in the question is inconsistent.]
We have  M2N2 (MT N)–1 (MN–1)T = M2N2N–1 (MT)–1 (N–1)T
MT
= M2 N (MT)–1 (N–1)T MT = –M2NM–1 N–1M
(∵ MT = –M, NT = –N and (N–1)T = (NT)–1
= – M (NM) (NM)–1 M             (∵ MN = NM)
= – MM = –M2

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 4

If the adjoint of a 3 x 3 matrix P is  then the possible value(s) of the determinant of P is (are) 

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 4

ANSWER :- a,d

Solution :-  |adj P| = |P|^2 as (adj(P)| = |P|^(n-1))

|adj P| = 1(3-7) -4(6-7) +4(2-1) = 4

Hence, |P| = 2 or -2

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 5

For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct?

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 5

(a)(N' M N)' = (M N)'N = N'M 'N = N'M N or –N'M N According as M  is symm. or skew symm. ∴correct
(b) (MN – NM)' = (MN)' – (NM)' = N'M' – M'N' = NM – MN = –(MN – NM)
∴ It is skew symm. Statement B is also correct.
(c)(MN)' = N'M' = NM ¹ MN
∴ Statement C is incorrect
(d) (adj M) (adj N) = adj (MN) is incorrect.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 6

Let ω be a complex cube root of unity with ω ≠ 1 and P = [pij] be a n × n matrix with pij = ωi+j. Then p2 ≠ 0, when n =

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 6

It shows P2 = 0 if n is a multiple of 3.
So for P2 ≠ 0, n should not be a multiple of 3 i.e. n can take values 55, 58, 56

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 7

Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 7

 where a, b, c are integers.

M is invertible if 


∴ (a) is not correct.
If [bc] = [ab] ⇒ b = a = c ⇒ ac = b2
∴ (b) is not correct.

∴ M is invertible.
(c) is correct
As ac ≠ (integer)2 ⇒ ac ≠ b2
∴ (d) is correct.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 8

Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N2 and M2 = N4, then

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 8

Given MN = NM, M ≠ N2 and M2 = N4.
Then M2 = N4 ⇒ (M + N2) (M – N 2) = 0
⇒ (i) M + N2 = 0 and M – N2 ≠ 0
(ii) |M + N2| = 0 and |M – N2| = 0
In each case |M + N2| = 0
∴ |M2 + MN2| = |M| |M + N2| = 0
∴ (a) is correct and (c) is not correct.
Also we know if |A| = 0, then there can be many matrices U, such that AU = 0
∴ (M2 + MN2)U = 0 will be true for many values of U.
Hence (b) is correct.
Again if AX = 0 and |A| = 0, then X can be non-zero.
∴ (d) is not correct.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 9

Which of the following values of a satisfy the equation

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 9



⇒ 2α2(–2α) = –324α  ⇒ α3 – 81α = 0 ⇒ α = 0, 9, –9

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 10

Let X and Y be two arbitrary, 3 × 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3, non zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 10

X ' = –X, Y ' = –Y, Z ' = Z
(Y3Z4 – Z4Y3)' = (Z4)'(Y3)' – (Y3)'(Z4)'
= (Z')4(Y')3 – (Y')3(Z')4
= –Z4Y3 + Y3Z4 = Y3Z4 – Z4Y3

∴ Symmetric matrix.
Similarly X44 + Y44 is symmetric matrix and X4Z3 – Z3X4 and X23 + Y23 are skew symmetric matrices.

*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 11

 Suppose Q = [qij] is a matrix such that PQ = kI, where  and I  is the  identity matrix of order 3. then

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 11

Comparing the third elements of 2nd row on both sides, we get


*Multiple options can be correct
Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 12

Let  Consider th e system of lin ear equations

ax + 2y = λ
3x – 2y = μ

Which of the following statement(s) is (are) correct?

Detailed Solution for Test: MCQs (One or More Correct Option): Matrices and Determinants | JEE Advanced - Question 12

ax + 2y = λ
3x – 2y = μ

For unique solution, 
∴ (b) is the correct option.
For infinite many solutions and a = – 3

∴ (c) is the correct option.

⇒ system has no solution.
⇒ (d) is the correct option.

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