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11.
If A, $open square brackets table row 2 cell negative 3 end cell row cell negative 4 end cell 1 end table close square brackets$ then adj (3A2 + 12A) is equal to

$open square brackets table row 72 cell negative 63 end cell row cell negative 84 end cell 51 end table close square brackets$
$open square brackets table row 72 cell negative 84 end cell row cell negative 63 end cell 51 end table close square brackets$
$open square brackets table row 51 63 row 84 72 end table close square brackets$
$open square brackets table row 51 63 row 84 72 end table close square brackets$

open square brackets table row 51 63 row 84 72 end table close square brackets

Given space straight A space equals space open square brackets table row 2 cell negative 3 end cell row cell negative 4 end cell 1 end table close square brackets 3 straight A squared space equals space open square brackets table row 16 cell negative 9 end cell row cell negative 12 end cell 13 end table close square brackets 12 space straight A space equals space open square brackets table row 24 cell negative 36 end cell row cell negative 48 end cell 12 end table close square brackets therefore space 3 straight A squared space plus space 12 space straight A space equals space open square brackets table row 72 cell negative 63 end cell row cell negative 84 end cell 51 end table close square brackets adj space left parenthesis 3 straight A squared space plus space 12 space straight A right parenthesis space equals space open square brackets table row 51 63 row 84 72 end table close square brackets

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12.

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr (A), the sum of diagonal entries of A. Assume that A₂= I.
Statement −1: If A ≠ I and A ≠ − I, then det A = − 1.
Statement −2: If A ≠ I and A ≠ − I, then tr (A) ≠ 0.

Statement −1 is false, Statement −2 is true
Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

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13.

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

If det A = ± 1, then A^–1 exists but all its entries are not necessarily integers
If detA ≠ ± 1, then A^–1 exists and all its entries are non-integers
If detA = ± 1, then A^–1 exists and all its entries are integers
If detA = ± 1, then A^–1 exists and all its entries are integers

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14.

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay. Then a² + b² + c² + 2abc is equal to

2
-1
0
0

391 Views

15.

If D = $open vertical bar table row 1 1 1 row 1 cell 1 plus straight x end cell 1 row 1 1 cell 1 plus straight y end cell end table close vertical bar$ for x ≠ 0, y ≠ 0 then D is

divisible by neither x nor y
divisible by both x and y
divisible by x but not y
divisible by x but not y

138 Views

16.

Let A = $open square brackets table row 5 cell 5 straight alpha end cell straight alpha row 0 straight alpha cell 5 straight alpha end cell row 0 0 5 end table close square brackets$ If |A²| = 25, then |α| equals

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17.

Let $straight A space equals space open parentheses table row 1 2 row 3 4 end table close parentheses space and space straight B space equals open parentheses table row straight a 0 row 0 straight b end table close parentheses$ a , b ∈ N. Then

there cannot exist any B such that AB = BA
there exist more than one but finite number of B’s such that AB = BA
there exists exactly one B such that AB = BA
there exists exactly one B such that AB = BA

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18.

If a₁, a₂, a₃,…, a_n,… are in G.P., then the determinant

$space increment space equals space open vertical bar table row cell log space straight a subscript straight n end cell cell log space straight a subscript straight n plus 1 end subscript end cell cell log space straight a subscript straight n plus 2 end subscript end cell row cell log space straight a subscript straight n plus 3 end subscript end cell cell log space straight a subscript straight n plus 4 end subscript end cell cell log space straight a subscript straight n plus 5 end subscript end cell row cell log space straight a subscript straight n plus 6 end subscript end cell cell log space straight a subscript straight n plus 7 end subscript end cell cell log space straight a subscript straight n plus 8 end subscript end cell end table close vertical bar space is space equal space to$

480 Views

19.

If the system of linear equations
x + ky + 3z = 0
3x + ky – 2z = 0
2x + 4y – 3z = 0

has a non-zero solution (x,y,z), then xz/y² is equal to

20.

If $|\begin{matrix} x - 4 & 2 x & 2 x \\ 2 x & x - 4 & 2 x \\ 2 x & 2 x & x - 4 \end{matrix}| = (A + B x) (x - A)^{2}$ , then the ordered pair (A,B) is equal to