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D Divya Prakash Singh
Given determinant  and given that a, b, c are in A.P. That means , 2b =a+c Applying the row transformations,    and then  we have; Now, applying another row transformation, , we have Clearly we have the determinant value equal to zero; Hence the option (A) is correct.    

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D Divya Prakash Singh
We have a system of equations;           So, we will convert the given system of equations in a simple form to solve the problem by the matrix method; Let us take, ,  Then we have the equations;          We can write it in the matrix form as  , where Now, Finding the determinant value of A;                 Hence we can say that A is non-singular  its invers exists; Finding cofactors of...

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D Divya Prakash Singh
Given determinant  Multiplying the first column by  and the second column by , and expanding the third column, we get Applying column transformation,  we have then; Here we can see that two columns  are identical. The determinant value is equal to zero.  Hence proved.

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D Divya Prakash Singh
Given determinant  Applying the row transformation;   and  we have then; Now, applying another row transformation  we have; We can expand the remaining determinant along , we have; Hence the result is proved.  

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D Divya Prakash Singh
Given determinant  Applying the column transformation,  we have then; Taking common factor (a+b+c) out from the column first; Applying   and  , we have then; Now we can expand the remaining determinant along  we have;   Hence proved.  

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D Divya Prakash Singh
Given the determinant  Applying the row transformations;   and  then we have; Applying row transformation  we have then; Now we can expand the remaining determinant to get the result; hence the given result is proved.

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D Divya Prakash Singh
Given determinant   Applying Row transformations; and  , then we have; Expanding the remaining determinant; hence the given result is proved.

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D Divya Prakash Singh
We have determinant   Applying row transformations;   , we have then; Taking out the common factor 2(x+y) from the row first. Now, applying the column transformation;  and   we have ; Expanding the remaining determinant; .        

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D Divya Prakash Singh
We have determinant   Applying row transformations;    and  then  we have then; Taking out the common factor -y from the row first. Expanding the remaining determinant;      

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D Divya Prakash Singh
Given that ; So, let us assume that   Hence its inverse exists;     or  ; so, we now calculate the value of  Cofactors of A;                                               Finding the inverse of B ; Hence its inverse exists; Now, finding the ;                                                                                           Hence proved L.H.S. =R.H.S..    

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D Divya Prakash Singh
Given that ; So, let us assume that  matrix and  then; Hence its inverse exists;     or  ; so, we now calculate the value of  Cofactors of A;                                              ??????? ??????? Finding the inverse of C; Hence its inverse exists; Now, finding the ;                                                                                            or  Now, finding the...

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D Divya Prakash Singh
We know from the identity that; . Then we can find easily,  Given   and   Then we have to basically find the  matrix.   So, Given matrix  Hence its inverse  exists; Now, as we know that So, calculating cofactors of B,                                                                                   Now, We have both  as well as  ; Putting in the relation we know;           

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D Divya Prakash Singh
Given matrix  Taking common factors a,b and c from the column  respectively. we have;  Applying , we have; Then applying  , we get; Applying , we have; Now, applying column transformation; , we have So we can now expand the remaining determinant along  we have; Hence proved.

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D Divya Prakash Singh
Given determinant  Applying the row transformation;  we have; Taking common factor (3x+a) out from first row. Now applying the column transformations;  and . we get;            as    , or   or   

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D Divya Prakash Singh
We have given  Applying the row transformations;  we have; Taking out common factor 2(a+b+c) from the first row; Now, applying the column transformations;  we have; and given that the determinant is equal to zero. i.e., ;  So, either  or . we can write  as;  are non-negative. Hence . we get then  Therefore, if given  = 0 then either  or .      

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D Divya Prakash Singh
Given determinant ; .  

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D Divya Prakash Singh
We have the   Multiplying rows with a, b, and c respectively. we get;                                   = R.H.S. Hence proved. L.H.S. =R.H.S.

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D Divya Prakash Singh
Calculating the determinant value of ; Clearly, the determinant is independent of .
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