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  • Help me understand! - Matrices and Determinants - JEE Main-4

Let for i=1,2,3,p_{i}(x)  be a polynomial of degree 2 in  x,p'_{i}(x)\: and \: p"_{i}(x)

be the first and second order derivatives of  p_{i}(x) respectively. Let,

and B(x)=\left [ A(x) \right ]^{T}A\left ( x \right ) Then determinant of B(x):

  • Option 1)

      is a polynomial of degree 6 in x.
     

  • Option 2)

      is a polynomial of degree 3 in x. 

  • Option 3)

    is a polynomial of degree 2 in x.

     

  • Option 4)

     does not depend on x.

 

Answers (1)

best_answer

As we have learned

Summation of two determinants -

-

 

 We have A(x) = \begin{bmatrix} a_1x^2+b_1x+c_1 & 2a_1x+b_1 &2a_1 \\ a_2x^2b_2x+c_2 & 2a_2x+b_2&2a_2 \\ a_3x^2+b_3x+c_3& 2a_3x+b_3 & 2a_3 \end{bmatrix}

 

|B(x)|= |(A(x))^T(A(x))|

= |(A(x))^T||(A(x))|

= |(A(x))||(A(x))|

= |(A(x))|^2

On applying addition of determinants we get sum of 6 determinants out of which five are 'O' and |A(x)|\Rightarrow \begin{vmatrix} c_1 & b_1 &a_1 \\ c_2 & b_2 &a_2 \\ c_3 &b_3 & a_3 \end{vmatrix}

 

 

 

 

 

 


Option 1)

  is a polynomial of degree 6 in x.
 

Option 2)

  is a polynomial of degree 3 in x. 

Option 3)

is a polynomial of degree 2 in x.

 

Option 4)

 does not depend on x.

Posted by

gaurav

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