Matrices & Determinants
Determinant Properties
Grade Class 12
Question:
The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|
(A) a + b + c
(B) (a + b) (b + c) (c + a)
(C) a<sup>2</sup> + b<sup>2</sup> + c<sup>2</sup>
(D) (a - b)(b - c) (c - a)
Step-by-Step Solution
Key Concept: Perform row operations to simplify the determinant and factorize it.
Let the determinant be \Delta. Applying C2 -> C2 + C1, we get \Delta = |a^2 a^2-(b-c)^2+a^2 bc; b^2 b^2-(c-a)^2+b^2 ca; c^2 c^2-(a-b)^2+c^2 ab|. This can be simplified further to show factors like (a+b+c), (a^2+b^2+c^2), and (a-b)(b-c)(c-a).
Correct Answer: A,C,D