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NCERT Class 12 · Ch 4 Miscellaneous Exercise · Q6

NCERT Class 12 Maths · Chapter 4 Determinants · Miscellaneous Exercise Question 6

Evaluate the determinant of \(\left[\begin{matrix}1 & x & y\\1 & x + y & y\\1 & x & x + y\end{matrix}\right]\).

💡 Concept Hints

  1. Identify whether this needs determinant expansion, an inverse, or the absolute determinant formula for area.
  2. For a determinant use signed cofactors. For a system first check invertibility; for an area take half the absolute determinant.

✅ Step-by-Step Solution

Final Answer: \(x y\)
  1. Expand the determinant along row 1, using cofactors Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ.
  2. The cofactors in this row are \(\left[\begin{matrix}x^{2} + x y + y^{2} & - x & - y\end{matrix}\right]\).
  3. The expansion terms are \(x^{2} + x y + y^{2}\), \(- x^{2}\), \(- y^{2}\).
  4. Adding and simplifying gives det A = \(x y\).

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