Determinants

Let A= bmatrix & 1 & 2\ 2 & 3 & 0\ 0 & 4 & 5 bmatrix and B= bmatrix1 & 0 & 0\ 0 & -5 & 0\ 0 & 4 & -2 bmatrix+adj(A). If (B)=66, then (adj(A)) equals :

1. 289
2. 361
3. 441
4. 529

Let A= bmatrix-1 & 1 & -1\ 1 & 0 & 1\ 0 & 0 & 1 bmatrix satisfy A2+ (adj (adj(A) ) )+ (adj(A) (adj (adj(A) ) ) )= bmatrix2 & -2 & 2\ -2 & 0 & -1\ 0 & 0 & -1 bmatrix for some , R. Then ( - )2 is equal to .

Let A be a 3 3 matrix such that AT bmatrix1\ 0\ 1 bmatrix= bmatrix5\ 2\ 2 bmatrix, AT bmatrix0\ 0\ 1 bmatrix= bmatrix3\ 1\ 1 bmatrix, A bmatrix1\ 0\ 1 bmatrix= bmatrix3\ 4\ 4 bmatrix and A bmatrix0\ 0\ 1 bmatrix= bmatrix1\ 3\ 1 bmatrix. If (A)=1, then (adj (A2+A ) ) is equal to:

1. 16
2. 25
3. 49
4. 64

Let A= bmatrix1 & 1 & 2\ -2 & 0 & 1\ 1 & 3 & 5 bmatrix. Then the sum of all elements of the matrix adj (adj (2(adj A)-1 ) ) is equal to:

1. 3
2. 4
3. -4
4. -3

Let P = [pij] and Q = [qij] be two square matrices of order 3 such that qij = 2(i+j-1)pij and (Q) = 210. Then the value of ( adj( adj P)) is:

1. 81
2. 16
3. 124
4. 32

Let f(x) = 7x10 + 9x8(1 + x2 + 2x9)2 dx, x > 0, x 0 f(x) = 0 and f(1) = 14. If A = bmatrix 0 & 0 & 1 \ 14 & f'(1) & 1 \ 2 & 4 & 1 bmatrix and B = adj(adj A) be such that |B| = 81, then 2 is equal to

1. 2
2. 4
3. 3
4. 1

Let |A| = 6, where A is a 3 3 matrix. If |adj (3 adj (A2 adj(2A) ) ) | = 2m 3n, m, n N, then m + n is equal to .

Let A be a 3 3 matrix such that A + AT = O. If A bmatrix 1 \ -1 \ 0 bmatrix = bmatrix 3 \ 3 \ 2 bmatrix, A2 bmatrix 1 \ -1 \ 0 bmatrix = bmatrix -3 \ 19 \ -24 bmatrix and (adj(2 adj(A + I))) = (2) (3) (11) , , , are non-negative integers, then + + is equal to

Let A = bmatrix 2 & 2+p & 2+p+q \ 4 & 6+2p & 8+3p+2q \ 6 & 12+3p & 20+6p+3q bmatrix. If (adj(adj(3A))) = 2m 3n, m, n N, then m + n is equal to

1. 22
2. 20
3. 24
4. 26

Let A be a 3 3 matrix such that |adj(adj(adj A))| = 81. If S = \n Z : (|adj(adj A)| ) (n-1)22 = |A| (3n2 - 5n - 4 ) \, then n S |A (n2 + n ) | is equal to :

1. 820
2. 866
3. 750
4. 732

Let A be a matrix of order 3 3 and |A| = 5. If |2 adj(3A adj(2A))| = 2 3 5 , , , N, then + + is equal to

1. 26
2. 27
3. 25
4. 28

Let I be the identity matrix of order 3 3 and for the matrix A = bmatrix & 2 & 3 \ 4 & 5 & 6 \ 7 & -1 & 2 bmatrix, |A| = -1. Let B be the inverse of the matrix adj (A adj (A2 ) ). Then |( B + I)| is equal to

Let a R and A be a matrix of order 3 3 such that (A) = -4 and A + I = bmatrix 1 & a & 1 \ 2 & 1 & 0 \ a & 1 & 2 bmatrix, where I is the identity matrix of order 3 3. If ((a + 1) adj((a - 1)A)) is 2m 3n, m, n \0, 1, 2, , 20\, then m + n is equal to :

1. 14
2. 17
3. 15
4. 16

Let A = [aij] = bmatrix 5 128 & 4 5 \ 5 8 & 4 25 bmatrix. If Aij is the cofactor of aij, Cij = k=12 aikAjk, 1 i, j 2, and C = [Cij], then 8|C| is equal to :

1. 288
2. 262
3. 222
4. 242