If A =  and A adj A = AAT, then 5a +b is equal to from Mathem

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 Multiple Choice QuestionsMultiple Choice Questions

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1.

If A = open square brackets table row cell 5 straight a end cell cell negative straight b end cell row 3 2 end table close square brackets and A adj A = AAT, then 5a +b is equal to

  • -1

  • 5

  • 4

  • 4


B.

5

Given, A =open square brackets table row cell 5 straight a end cell cell negative straight b end cell row 3 2 end table close square brackets and A adj A = AAT, Clearly, A (adj A) = |A|In|
space equals space open vertical bar table row cell 5 straight a end cell cell negative straight b end cell row 3 2 end table close vertical bar straight I subscript 2
space equals space left parenthesis 10 straight a space plus space 3 straight b right parenthesis straight I subscript 2 space equals space left parenthesis 10 straight a space plus space 3 straight b right parenthesis open square brackets table row 1 0 row 0 1 end table close square brackets
equals open square brackets table row cell 10 straight a plus 3 straight b end cell 0 row 0 cell 10 straight a plus 3 straight b end cell end table close square brackets space.... space left parenthesis straight i right parenthesis
and space AA to the power of straight T space equals space open square brackets table row cell 5 straight a end cell cell negative straight b end cell row 3 2 end table close square brackets open square brackets table row cell 5 straight a end cell 3 row cell negative straight b end cell 2 end table close square brackets
equals open square brackets table row cell 25 straight a squared plus straight b squared end cell cell 15 straight a minus 2 straight b end cell row cell 15 straight a minus 2 straight b end cell 13 end table close square brackets space... space left parenthesis ii right parenthesis
because space straight A space left parenthesis adj space straight A right parenthesis space equals space AA to the power of straight T
therefore space open square brackets table row cell 10 straight a plus 3 straight b end cell 0 row 0 cell 10 straight a plus 3 straight b end cell end table close square brackets space equals open square brackets table row cell 25 straight a squared plus straight b squared end cell cell 15 straight a minus 2 straight b end cell row cell 15 straight a minus 2 straight b end cell 13 end table close square brackets space
using space eqs space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis
rightwards double arrow space 15 straight a minus space 2 straight b equals space 0 space
rightwards double arrow space fraction numerator 2 straight b over denominator 15 end fraction space space space.. left parenthesis iii right parenthesis
and space 10 space straight a space plus space 3 straight b space equals space 13
On space substituting space the space value space of space apostrophe straight a apostrophe space from space Eq. space left parenthesis iii right parenthesis space in space eq space left parenthesis iv right parenthesis comma space we space get
10. open parentheses fraction numerator 2 straight b over denominator 15 end fraction close parentheses plus 3 straight b space equals space 13
rightwards double arrow space fraction numerator 20 straight b space plus space 45 straight b over denominator 15 end fraction space equals space 13 space rightwards double arrow space fraction numerator 65 straight b over denominator 15 end fraction space equals space 13

Now, substituting the value of b in Eq. (iii) we get 
5a = 2
Hence, 5a + b = 2 +3 = 5

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2.

The system of linear equations x+λy−z=0; λx−y−z=0; x+y−λz=0 has a non-trivial solution for

  • infinitely many values of λ.

  • exactly one value of λ.

  • exactly two values of λ.

  • exactly two values of λ.

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3.

A = open square brackets table row 1 2 2 row 2 1 cell negative 2 end cell row straight a 2 straight b end table close square brackets is a matrix satisfying the equation AAT = 9I, Where I is 3 x 3 identity matrix, then the ordered pair (a,b) is equal to

  • (2,-1)

  • (-2,1)

  • (2,1)

  • (2,1)

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4.

The set of all values of λ for which the system of linear equations 

2x1-2x2+x3 = λx1
2x1- 3x2 + 2x3 = λx2
-x1 + 2x2 = λx3
a non- trivial solution.

  • is an empty set

  • is a singleton set

  • contains two elements

  • contains two elements

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5.

If α, β ≠ 0 and f(n) = αn+ βn and 

open vertical bar table row 3 cell 1 plus straight f left parenthesis 1 right parenthesis space space space space end cell cell 1 plus space straight f left parenthesis 2 right parenthesis end cell row cell 1 plus straight f left parenthesis 1 right parenthesis space space space space space end cell cell 1 plus straight f left parenthesis 2 right parenthesis space space space space space space end cell cell 1 plus straight f left parenthesis 3 right parenthesis end cell row cell 1 plus straight f left parenthesis 2 right parenthesis space space space end cell cell 1 plus straight f left parenthesis 3 right parenthesis space space space space space space space end cell cell 1 plus space straight f left parenthesis 4 right parenthesis end cell end table close vertical bar
= K(1-α)2(1-β)2(α- β)2, then K is equal to 

  • αβ 

  • 1/αβ 

  • 1

  • 1

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6.

If A is a 3x3 non- singular matrix such that AAT = ATA, then BBT is equal to

  • l +B
  • l
  • B-1

  • B-1

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7.

Let P and Q be 3 × 3 matrices with P ≠ Q. If P3= Qand P2Q = Q2P, then determinant of(P2+ Q2) is equal to

  • -2

  • 1

  • 0

  • 0

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8.

The number of values of k for which the linear equations
4x + ky + 2z = 0
kx + 4y + z = 0
2x + 2y + z = 0
posses a non-zero solution is:

  • 3

  • 2

  • 1

  • 1

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9.

Consider the system of linear equation
x1 + 2x2 + x3 = 3
2x1 + 3x2 + x3 = 3
3x1 + 5x2 + 2x3 = 1
The system has

  • infinite number of solutions

  • exactly 3 solutions

  • a unique solution

  • a unique solution

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10.

Let A be a 2 × 2 matrix with non-zero entries and let A2 = I, where I is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.
Statement-1: Tr(A) = 0.
Statement-2: |A| = 1.

  • 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 false.

  • Statement-1 is true, Statement-2 is false.

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