NCERT Class 12 · Ch 4 Exercise 4.3 · Q4
NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.3 Question 4
Evaluate \(\left[\begin{matrix}1 & x & y z\\1 & y & x z\\1 & z & x y\end{matrix}\right]\) by expanding along the third column.
💡 Concept Hints
- Identify whether this needs determinant expansion, an inverse, or the absolute determinant formula for area.
- 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: \(- \left(x - y\right) \left(x - z\right) \left(y - z\right)\)
- The third-column cofactors, with signs +,−,+, are \(\left[\begin{matrix}- y + z\\x - z\\- x + y\end{matrix}\right]\).
- Multiply them by the corresponding third-column entries and add: \(- x^{2} y + x^{2} z + x y^{2} - x z^{2} - y^{2} z + y z^{2}\).
- Factor the polynomial to obtain \(- \left(x - y\right) \left(x - z\right) \left(y - z\right)\).
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