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Q : 16         Which of the following is correct

(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of these

The answer is (C) Determinant is a number associated to a square matrix. As we know that To every square matrix of order n, we can associate a number (real or complex) called determinant of the square matrix A, where element of A.

Q : 15        Let A be a square matrix of order  , then  is equal to

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

Assume a square matrix A of order of . Then we have; (Taking the common factors k from each row.)   Therefore correct option is (C).

Q : 14            By using properties of determinants, show that:

Given determinant:                               Let  Then we can clearly see that each column can be reduced by taking common factors like a,b, and c respectively from C1,C2,and C3. We then get; Now, applying column transformations:   and   then we have; Now, expanding the remaining determinant: . Hence proved.

Q : 13        By using properties of determinants, show that:

We have determinant:                              Applying row transformations,     and    then we have; taking common factor out of the determinant; Now expanding the remaining determinant we get; Hence proved.

Q : 12        By using properties of determinants, show that:

Give determinant   Applying column transformation  we get;       [after taking the (1+x+x2 ) factor common out.] Now, applying row transformations,     and then . we have now, As we know  Hence proved.

Q : 11        By using properties of determinants, show that:

Given determinant    Applying    we get; Taking 2(x+y+z) factor out, we get; Now, applying row transformations,   and then . we get; Hence proved.

Q : 11         By using properties of determinants, show that:

Given determinant:                                 We apply row transformation:  we have; Taking common factor (a+b+c) out. Now, applying column tranformation     and then   We have; Hence Proved.

Q : 10          By using properties of determinants, show that:

Given determinant:                                        Applying row transformation   we get;                             [taking common (3y + k) factor] Now, applying column transformation   and  Hence proved.

Q : 10        By using properties of determinants, show that:

Given determinant:                                        Applying row transformation:   then we have; Taking a common factor: 5x+4 Now, applying column transformations   and

Q : 9        By using properties of determinants, show that:

We have the determinant:                                             Applying the row transformations  and then  , we have; Now, applying ; we have Now, expanding the remaining determinant; Hence proved.

Q : 8        By using properties of determinants, show that:

Given determinant :                                           , Applying column transformation  and then  We get, Now, applying column transformation , we have: Hence proved.

Q : 8        By using properties of determinants, show that:

(i)

We have the determinant  Applying the row transformations   and then   we have: Now, applying  we have:     or     Hence proved.

Q : 7          Using the property of determinants and without expanding, prove that

Given determinant :  As we can easily take out the common factors a,b,c from rows  respectively. So, get then: Now, taking common factors a,b,c from the columns  respectively. Now, applying rows transformations    and then  we have; Expanding to get R.H.S.

Q : 6          Using the property of determinants and without expanding, prove that

We have given determinant   Applying transformation,  we have then, We can make the first row identical to the third row so, Taking another row transformation:  we have, So, determinant has two rows  identical. Hence .

Q : 5           Using the property of determinants and without expanding, prove that

Given determinant :    Splitting the third row; we get, . Then we have, On Applying row transformation    and then  ; we get,  Applying Rows exchange transformation    and   , we have: also  On applying rows transformation,  and then    and then   Then applying rows exchange transformation;    and then . we have then; So, we now calculate the sum =  Hence proved.

Q : 4           Using the property of determinants and without expanding, prove that

We have determinant:    Applying  we have then; So, here column 3 and column 1 are proportional. Therefore, .

Q : 3          Using the property of determinants and without expanding, prove that

Given determinant  So, we can split it in two addition determinants:      [ Here two columns are identical ] and     [ Here two columns are identical ]                   Therefore we have the value of determinant = 0.

Q : 2      Using the property of determinants and without expanding, prove that

Given determinant  Applying the rows addition    then we have; So, we have two rows  and  identical hence we can say that the value of determinant = 0  Therefore .

Q : 1        Using the property of determinants and without expanding, prove that

We can split it in manner like;   So, we know the identity that If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero. Clearly, expanded determinants have identical columns. Hence the sum is zero.
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