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Matrices · Question 6

JEE Advanced 2024 Paper 1MatchingHard📈 Topic in 17% papers+4−1
1

Let and be the distinct roots of the equation x2+x-1=0. Consider the set T=\1, , \. For a 3 3 matrix M= (aij )3 3, define Ri=ai1+ai2+ai3 and Cj=a1j+a2j+a3j for i=1,2,3 and j=1,2,3. Match each entry in List-I to the correct entry in List-II.

  1. (P) → (4) (Q) → (2) (R) → (5) (S) → (1)
  2. (P) → (2) (Q) → (4) (R) → (1) (S) → (5)
  3. (P) → (2) (Q) → (4) (R) → (3) (S) → (5)
  4. (P) → (1) (Q) → (5) (R) → (3) (S) → (4)

Answer

(3) (P) → (2) (Q) → (4) (R) → (3) (S) → (5)

Solution

+ =-1 1+ + =0. With 0.618, -1.618, the only way three entries of T add to 0 is \1, , \ (each once).

(P) Every row and column is a permutation of 1, , : Latin squares of order 3 =12. P → 2

(Q) Symmetric with column sums 0: each symbol occurs once on the diagonal (off-diagonal entries come in pairs); diagonal in 3!=6 ways fixes the rest, so count =6. Q → 4

(R) M j=(a12,0,-a23)T, so a solution exists; a 3 3 skew-symmetric matrix is singular, hence infinitely many solutions. R → 3

(S) M(1,1,1)T=0 |M|=0. S → 5

Answer: (C)

JEE Advanced 2024 Paper 1

Keep practising — Matrices

2

Which one of the following matrices can be obtained by performing elementary row transformations on the 3 3 identity matrix ?

  1. bmatrix 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 bmatrix
  2. bmatrix 1 & 1 & 1 \ 2 & 3 & 4 \ 1 & 2 & 1 bmatrix
  3. bmatrix 1 & 1 & 1 \ 2 & 3 & 4 \ 2 & 5 & 8 bmatrix
  4. bmatrix 1 & 1 & 1 \ -1 & 1 & 2 \ 0 & 2 & 3 bmatrix
JEE Advanced 2026 Paper 1
3

Consider the matrix M = bmatrix 2 & -1 \ 1 & 0 bmatrix . Let p, q, r, s, a, b, c and d be integers such that M26 = bmatrix p & q \ r & s bmatrix and k=126 Mk = bmatrix a & b \ c & d bmatrix . Then which of the following statements is (are) TRUE ?

  1. There exists a 2 2 invertible matrix N with real entries such that MN = N bmatrix 1 & 1 \ 0 & 1 bmatrix
  2. The value of a is 378
  3. For any two given integers m and n, there exist unique integers x and y such that px + qy = m and rx + sy = n
  4. For each positive real number t, the system of linear equations (a + t)x + by = 1 cx + (d + t)y = -1 has a unique solution
JEE Advanced 2026 Paper 1
4

Let R denote the set of all real numbers and let i= -1. Consider the matrices S= bmatrix 0 & -1 \ 1 & 0 bmatrix and T= bmatrix 1 & 1 \ 0 & 1 bmatrix. Let a,b,c,d be real numbers such that ST= bmatrix a & b \ c & d bmatrix. Let H=\x+iy: x,y R and y>0\. Then which of the following statements is (are) TRUE ?

  1. b+iad+ic=i
  2. If = -1+i 32, then a +bc +d=
  3. If m is an integer greater than 2 such that (ST)2=(ST)m, then m is an integer multiple of 8
  4. If z H, then az+bcz+d H
JEE Advanced 2026 Paper 2
5

Consider the matrix P = pmatrix 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 pmatrix. Let the transpose of a matrix X be denoted by XT. Then the number of 3 3 invertible matrices Q with integer entries, such that Q-1 = QT and PQ = QP, is

  1. 32
  2. 8
  3. 16
  4. 24
JEE Advanced 2025 Paper 1
6

Let I= pmatrix1 & 0\ 0 & 1 pmatrix and P= pmatrix2 & 0\ 0 & 3 pmatrix. Let Q= pmatrixx & y\ z & 4 pmatrix for some non-zero real numbers x, y, and z, for which there is a 2 2 matrix R with all entries being non-zero real numbers, such that QR=RP. Then which of the following statements is (are) TRUE?

  1. The determinant of Q-2I is zero
  2. The determinant of Q-6I is 12
  3. The determinant of Q-3I is 15
  4. yz=2
JEE Advanced 2025 Paper 2

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