Determinants

More Topic from Mathematics

Question 1

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)

Solution

D.

(-2,-1)

Given, 
straight A space equals space 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
straight A to the power of straight T space equals space open square brackets table row 1 2 straight a row 2 1 cell negative 2 end cell row 2 cell negative 2 end cell straight b end table close square brackets
AA to the power of straight T space equals space 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 open square brackets table row 1 2 straight a row 2 1 cell negative 2 end cell row 2 cell negative 2 end cell straight b end table close square brackets
equals space open square brackets table row 9 0 cell straight a plus 4 plus 2 straight b end cell row 0 9 cell space 2 straight a plus 2 minus 2 straight b end cell row cell straight a plus 4 plus 2 straight b end cell cell space 2 straight a plus 2 minus 2 straight b end cell cell space straight a squared plus 4 plus straight b squared end cell end table close square brackets.
It is given that,
open square brackets table row 9 0 cell space space space straight a plus 4 plus 2 straight b end cell row 0 9 cell space space space space 2 straight a plus 2 minus 2 straight b end cell row cell straight a plus 4 plus 2 straight b end cell cell space space 2 straight a plus 2 minus 2 straight b end cell cell space space space space straight a squared plus 4 plus straight b squared end cell end table close square brackets space equals space 9 open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets

rightwards double arrow
open square brackets table row 9 0 cell space space space straight a plus 4 plus 2 straight b end cell row 0 9 cell space space space space 2 straight a plus 2 minus 2 straight b end cell row cell straight a plus 4 plus 2 straight b end cell cell space space 2 straight a plus 2 minus 2 straight b end cell cell space space space space straight a squared plus 4 plus straight b squared end cell end table close square brackets space equals space 9 open square brackets table row 9 0 0 row 0 9 0 row 0 0 9 end table close square brackets
On comparing we get,
a+ 4 +2b = 0
a+ 2b = -4   ... (i)
2a + 2-2b = 0
a-b= -1    ... (ii)
a2 + 4 +b2 = 9  ... (iii)
On solving eqs. (i) and (ii) we get
a = - 2, b = - 1
Hence, (a,b) ≡ (-2,-1)

Question 2

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

  • no solution

Solution

D.

no solution

Subtracting the Eq. (ii) – Eq. (i)
We get x1 + x2 = 0
Subtract equations
Eq. (iii) – 2 × eq. (ii)
x1 + x2 = 5

Therefore, no solutions

Question 3

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

  • 5

Solution

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

Question 4

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

  • l +B
  • l
  • B-1

  • (B-1)T

Solution

B.

l

If A is non - singular matrix then |A| ≠0
AAT = ATA and B = A-1AT
BBT = (A-1AT)(A-1AT)T
= A-1ATA(A-1)T       [∵ (AB)T= BTAT]
=A-1AAT(A-1)T        [∵ AAT = ATA]
=AT(A-1)T              [ ∵A-1A = l]
=A-1A)T                 [∵ (AB)T = BTAT]
lTl

Question 5

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 84 row 63 72 end table close square brackets

Solution

C.

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