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Let M be any 3 × 3 matrix with entires from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements MTM is seven is____________.

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    Answered by

    alok kumar singh | Contributor-Level 10

    a month ago

    M = [ a 1 a 2 a 3 b 1 b 2 c 3 c 1 c 2 c 3 ]    

    M T M = [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ] [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ]

    T r ( M T M ) = a 1 2 + b 1 2 + c 1 2 + a 2 2 + b 2 2 + c 2 2 + a 3 2 + b 3 2 + c 3 2 = 7           

    all   a i , b i , c i ∈ { 0 , 1 , 2 } f o r     i = 1, 2, 3

    Case 1 7 one’s and two zeroes which can occur in ways

    Case 2 One 2 three 1’s five zeroes =

     

    total such matrices = 504 + 36 = 540

     

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If the matrix A = [ 1 0 0 0 2 0 3 0 − 1 ]  satisfies the equation A20 = aA19 + βA = [ 1 0 0 0 4 0 0 0 1 ] for some real numbers a and b, then b - a is equal to…………
A
alok kumar singh

  A 2 = [ 1 0 0 0 2 0 3 0 − 1 ] [ 1 0 0 0 2 0 3 0 − 1 ] = [ 1 0 0 0 2 2 0 0 0 1 ]

A 3 = [ 1 0 0 0 2 2 0 0 0 1 ] [ 1 0 0 0 2 0 3 0 − 1 ] = [ 1 0 0 0 2 3 0 3 0 − 1 ]        

A 4 [ 1 0 0 0 2 3 0 3 0 − 1 ] [ 1 0 0 0 2 0 3 0 − 1 ] = [ 1 0 0 0 2 4 0 0 0 1 ]          

Similarly we get A19 =   = [ 1 0 0 0 2 1 9 0 3 0 − 1 ] & A 2 0 = [ 1 0 0 0 2 2 0 0 0 0 1 ]

=   [ 1 0 0 0 4 0 0 0 1 ]

⇒ 1 + α + β = 1 g i v e s     α + β = 0 . . . . . . . . ( i )        

  2 2 0 + ( 2 1 9 − 2 ) α = 4 f r o m ( i )

⇒ α = 4 − 2 2 0 2 1 9 − 2 = 4 ( 1 − 2 1 8 ) − 2 ( 1 − 2 1 8 ) = − 2          

So, b = 2

Hence b - a = 4

 

Consider the following system of equations:

x + 2y – 3z = a                     

2x + 6y – 11z = b

x – 2y + 7z = c,

where a, b and c are real constants. Then the system of equations:
A
alok kumar singh

Given x + 2y – 3z = a

2x + 6y – 11z = b

x – 2y + 7z = c

Here    Δ = | 1 2 − 3 2 6 − 1 1 1 − 2 7 | = ( 4 2 − 2 2 ) − 2 ( 1 4 + 1 1 ) − 3 ( − 4 − 6 ) = 2 0 − 5 0 + 3 0 = 0                                                                                

For infinite solution 

20a – 8b – 4c = 0 Þ 5a = 2b + c

Let A = { n ∈ N : H . C . F . ( n , 4 5 ) = 1 } and Let B = { 2 k : k ∈ { 1 ,   2 , . . . . . . ,   1 0 0 } } . Then the sum of all the elements of   A ∪ B     i s . . . . . . . . . .  
R
Raj Pandey

Sum of all elements of A ∩ B = 2   [Sum of natural number upto 100 which are neither divisible by 3 nor by 5]

= 2 [ 1 0 0 × 1 0 1 2 − 3 ( 3 3 × 3 4 2 ) − 5 ( 2 0 × 2 1 2 ) + 1 5 ( 6 × 7 2 ) ]

= 10100 – 3366 – 2100 + 630

              = 5264

Let  A = ( 2 − 1 0 2 ) . If B = I − 5 ​ C 1 ( a d j A ) + 5 ​ C 2 ( a d j A ) 2 − . . . 5 ​ C 5 ( a d j A ) 5 ,  then the sum of all elements of the matrix B is
A
alok kumar singh

Kindly go through the solution

B = (I – adjA)5

Let M = [ 0 − α α 0 ] , where a is a non-zero real number an N = ∑ k = 1 4 9 M 2 k . If (I – M2)N = -2I, then the positive integral value of a is…………….
A
alok kumar singh

N = M 2 + M 4 + . . . . . + M 9 8

= ( − α 2     I ) + ( − α 2     I ) 2 + . . . . + ( − α 2     I ) 4 9

= I ( − α 2 + α 4 − α 6 + . . . . − α 9 8 )

N =  − I ( α 2 − α 4 + α 6 . . . . . . . + α 9 8 )

= − I α 2 ( 1 − ( − α 2 ) 4 9 ) 1 − ( − α 2 )

N =  − I   α 2 ( 1 + α 9 8 ) 1 + α 2  

Now  ( I − m 2 ) N = − 2 I

( I + α 2 I ) ( − I α 2 ⋅ ( 1 + α 9 8 ) 1 + α 2 = − 2 I  

-> a100 + a2 = 2

->a = ± 1

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