Matrices · Question 1

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

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

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

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

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)