Matrices · Question 1

Let M be a 3 3 matrix such that M bmatrix1\ 0\ 0 bmatrix= bmatrix1\ 2\ 3 bmatrix, M bmatrix0\ 1\ 0 bmatrix= bmatrix0\ 1\ 2 bmatrix and M bmatrix0\ 0\ 1 bmatrix= bmatrix-1\ 1\ 1 bmatrix. If M bmatrixx\ y\ z bmatrix= bmatrix1\ 7\ 11 bmatrix, then x+y+z equals :

1. 4
2. 5
3. 7
4. 11

Let S= \A= bmatrixa & b\ c & d bmatrix : a, b, c, d \0, 1, 2, 3, 4\ and A2-4 A+3I=0 \ be a set of 2 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is :

1. 20
2. 17
3. 21
4. 19

Let A= bmatrix1 & 2\ 1 & bmatrix and B= bmatrix3 & 3\ & 2 bmatrix. If A2-4A+I=0 and B2-5B-6I=0, then among the two statements: (S1) : [(B-A)(B+A) ]T= bmatrix13 & 15\ 7 & 10 bmatrix and (S2) : (adj(A+B))=-5

1. only (S1) is correct
2. only (S2) is correct
3. both (S1) and (S2) are correct
4. both (S1) and (S2) are wrong

Consider the matrices A= bmatrix2 & -2\ 4 & -2 bmatrix and B= bmatrix3 & 9\ 1 & 3 bmatrix. If matrices P and Q are such that PA=B and AQ=B, then the absolute value of the sum of the diagonal elements of 2(P+Q) is .

Let A, B and C be three 2 × 2 matrices with real entries such that B = (I + A)-1 and A + C = I. If BC = bmatrix 1 & -5 \ -1 & 2 bmatrix and CB bmatrix x1 \ x2 bmatrix = bmatrix 12 \ -6 bmatrix, then x1 + x2 is

1. 4
2. 2
3. 0
4. −2

Let A = bmatrix 3 & -4 \ 1 & -1 bmatrix and B be two matrices such that A100 = 100B + I. Then the sum of all the elements of B100 is

The number of 3 2 matrices A, which can be formed using the elements of the set \-2, -1, 0, 1, 2\ such that the sum of all the diagonal elements of ATA is 5, is

Let A = bmatrix 0 & 2 & -3 \ -2 & 0 & 1 \ 3 & -1 & 0 bmatrix and B be a matrix such that B(I - A) = I + A. Then the sum of the diagonal elements of BTB is equal to

If A = bmatrix 2 & 3 \ 3 & 5 bmatrix, then the determinant of the matrix (A2025 - 3 A2024 + A2023 ) is

1. 12
2. 24
3. 28
4. 16

For some , R, let A = bmatrix & 2 \ 1 & 2 bmatrix and B = bmatrix 1 & 1 \ 1 & bmatrix be such that A2 - 4A + 2I = B2 - 3B + I = O. Then ( ( adj(A3 - B3)))2 is equal to .