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need solution for RD Sharma maths class 12 chapter 5 Determinants exercise Fill in the blanks question 15

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Answer: |C|=|A|^{2}=2^{2}=4

Hint: Here, we use basic concept of diagonal matrix and determinant of matrix

Given:  \begin{aligned} &A=\left[c_{i j}\right]_{3 \times 3}|A|=2 \text { and } c_{i j} \text { be cofactor of } a_{i j}\\ &C=\left[c_{i j}\right] \end{aligned}

Solution: c is cofactor matrix

                \begin{aligned} &A d j A=(C)^{T}\\ &\text { But Adj } A=\text { Det. of } C \end{aligned}

Because both are transpose of each other,

                |\operatorname{adj} A|=|C|=|A|^{n-1}

                Here n = 3 because A is 3\times 3 matrix,

                \begin{aligned} &|C|=|A|^{3-1} \\ &|C|=|A|^{2} \end{aligned}


                \begin{aligned} &|A|=2 \\ &|C|=(2)^{2}=4 \end{aligned}

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