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NCERT Class 12 · Ch 4 Miscellaneous Exercise · Q2

NCERT Class 12 Maths · Chapter 4 Determinants · Miscellaneous Exercise Question 2

Evaluate the determinant \(\left[\begin{matrix}\cos{\left(\alpha \right)} \cos{\left(\beta \right)} & \sin{\left(\beta \right)} \cos{\left(\alpha \right)} & - \sin{\left(\alpha \right)}\\- \sin{\left(\beta \right)} & \cos{\left(\beta \right)} & 0\\\sin{\left(\alpha \right)} \cos{\left(\beta \right)} & \sin{\left(\alpha \right)} \sin{\left(\beta \right)} & \cos{\left(\alpha \right)}\end{matrix}\right]\).

💡 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: \(1\)
  1. Expand along row 2, whose last entry is zero.
  2. The first two contributions simplify to sin²β(cos²α+sin²α) and cos²β(cos²α+sin²α).
  3. Both bracketed factors are 1; the sum is sin²β+cos²β=1.

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