Important Questions of Determinants Mathematics | Zigya

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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 bracketsthen 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
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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

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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 barfor 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

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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

  • 52

  • 1

  • 1/5

  • 1/5

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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,…, an,… 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

  • 1

  • 0

  • 4

  • 4

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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

  • 30

  • -10

  • 10

  • -30


20.

If x-42x2x2xx-42x2x2xx-4 = (A +Bx)(x-A)2, then the ordered pair (A,B) is equal to

  • (4,5)

  • (-4,-5)

  • (-4,3)

  • (-4,5)


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