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in Determinants by (115k points)
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The value of the determinant

\(\left| {\begin{array}{*{20}{c}} { - {a^2}}&{ab}&{ac}\\ {ab\;}&{ - {b^2}}&{bc}\\ {ac}&{bc}&{ - {c^2}} \end{array}} \right|\)

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Best answer
Correct Answer - Option 3 : 4a2 b2 c2

Given:

\(\left| {\begin{array}{*{20}{c}} { - {a^2}}&{ab}&{ac}\\ {ab\;}&{ - {b^2}}&{bc}\\ {ac}&{bc}&{ - {c^2}} \end{array}} \right|\)

Calculation:

Then determinant of given matrix is,

-a2(b2c2 – b2c2) – ab(-abc2 - abc2) + ac(ab2c + ab2c)

0 – ab(-2abc2) + ac(2ab2c)

2 a2b2c2 + 2 a2b2c2

4 a2b2c2

Properties of Determinants:

  • The value of the determinant remains unchanged if both rows and columns are interchanged.
  • If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
  • If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero.
  • If some or all elements of a row or column of a determinant are expressed as the sum of two (or more) terms, then the determinant can be            expressed as the sum of two (or more) determinants.
  • det (AT) = det (A)
  • det (kA) = kndet(A)
  • det(A.B) = det(A).det(B)

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