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JEE Main 2019
Matrices & Determinants
Matrices and Determinants
Hard

Question

Let a,b,ca, b, c be such that b(a+c)≠0b\left( {a + c} \right) \ne 0 if \left| {\matrix{ a & {a + 1} & {a - 1} \cr { - b} & {b + 1} & {b - 1} \cr c & {c - 1} & {c + 1} \cr } } \right| + \left| {\matrix{ {a + 1} & {b + 1} & {c - 1} \cr {a - 1} & {b - 1} & {c + 1} \cr {{{\left( { - 1} \right)}^{n + 2}}a} & {{{\left( { - 1} \right)}^{n + 1}}b} & {{{\left( { - 1} \right)}^n}c} \cr } } \right| = 0 then the value of nn :

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Solution

This problem involves simplifying an equation containing two determinants and then using their properties to find the value of nn. The key concepts we will use are the fundamental properties of determinants:

Key Concepts and Properties of Determinants:

  1. Transpose: The value of a determinant remains unchanged if its rows and columns are interchanged. That is, det⁡(A)=det⁡(AT)\det(A) = \det(A^T).
  2. Row/Column Interchange: If any two rows or columns of a determinant are interchanged, the sign of the determinant changes.
  3. Scalar Multiplication: If all elements of a row or column of a determinant are multiplied by a scalar kk, then the value of the determinant is multiplied by kk.
  4. Row/Column Operations: If
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