Find the determinant of the matrix $[0 p - q p - r q - p 0 q - r r - p r - q 0]$ .

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(p - q) (q - r) (r - p)

pqr

4pqr

answer is B.

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Detailed Solution

The data given in the question is,
Let the determinant of matrix A =

[0 p - q p - r q - p 0 q - r r - p r - q 0]

To find the determinant of the matrix A,
Taking transpose for the above given matrix we get,
|A| =

[0 q - p r - p p - q 0 r - q p - r q - r 0]

Taking (−1) from all the columns of the above we get,

|A| = \frac{1}{{(- 1)}^{3}} [0 q - p r - p p - q 0 r - q p - r q - r 0]

By solving the above equation, we get,
|A|=(−1)

[0 q - p r - p p - q 0 r - q p - r q - r 0]

Then the determinant of matrix A will be,
Δ=−|A|
The above can written as,
Δ=−Δ
While bringing the determinant on the one side we get,
Δ+Δ=0
2Δ=0
Δ=0
∴ The determinant of the above given matrix is 0 .
Hence, the determinant of the above given matrix is zero.

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