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Q : 19        Let      where .  Then

(A)                         (B)

(C)                  (D)

Given determinant  Now, given the range of  from  Therefore the  . Hence the correct answer is D.

Q : 18         If  x, y, z are nonzero real numbers, then the inverse of matrix     is

Given Matrix , As we know,  So, we  will find the , Determining its cofactor first,                Hence  Therefore the correct answer is (A).

Q : 17     If    are in A.P, then the determinant
is

(A)             (B)            (C)         (D)

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.

Q : 16         Solve the system of equations

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...

Q : 15        Using properties  of determinants, prove that

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.

Q :14        Using properties  of determinants, prove that

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.

Q : 13        Using properties  of determinants, prove that

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.

Q : 12        Using properties  of determinants, prove that

where p is any scalar.

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.

Q : 11        Using properties  of determinants, prove that

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

Q : 9        Evaluate

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; .

Q : 10        Evaluate

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;

Q : 8        Let , Verify that

(ii)

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..

Q : 8        Let . Verify that,

(i)

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...

Q : 7        If       and , find .

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;

Q : 6        Prove that    .

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.

Q : 5        Solve the equation

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

Q : 4        If and  are real numbers, and

Show that either  or

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 .

Q : 3        Evaluate    .

Given determinant ; .

Q : 2         Without expanding the determinant, prove that

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

Q : 1         Prove that the determinant    is independent of .

Calculating the determinant value of ; Clearly, the determinant is independent of .
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