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Provide solution for RD Sharma maths class 12 chapter Determinants exercise multiple choise question 10

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Solve given determinant


        Here\; \begin{vmatrix} -2a &a+b &a+c \\ b+a &-2b &b+c \\ c+a &c+b &-2c \end{vmatrix} is \; given

The three factor are given, we have to find another one factor.


        \begin{vmatrix} -2a &a+b &a+c \\ b+a &-2b &b+c \\ c+a &c+b &-2c \end{vmatrix}

Appling C1→C1+C2 , we get

        =\begin{vmatrix} b-a &a+b &a+c \\ a-b &-2b &b+c \\ a+b+2c &b+c &-2c \end{vmatrix}

Appling C2→C2+C3 , we get

        =\left | b-a2a+b+ca+ca-bc-bb+ca+b+2cb-c-2c \right |

Appling R3→R3+R2 , we get

        =\begin{vmatrix} b-a &2a+b+a &a+c \\ a-b &c-b &b+c \\ 2(c+a) &0 &b-c \end{vmatrix}

Appling R2→R2+R1 , we get

        =\begin{vmatrix} b-a &2a+b+a &a+c \\ 0 &2(a+c) &a+b+2c \\ 2(c+a) &0 &b-c \end{vmatrix}

Expanding along C1 , we get







Hence another factor of the given determinant is 4.

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Gurleen Kaur

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