Correct Answer - Option 3 : 4a
2 b
2 c
2
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)