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JEE Main 2022
Matrices & Determinants
Matrices and Determinants
Easy

Question

Let A={aij}A = \{ {a_{ij}}\} be a 3 ×\times 3 matrix, where {a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right. then det⁡(3Adj(2A−1))\det (3Adj(2{A^{ - 1}})) is equal to _____________.

Answer: 2

Solution

Key Concepts and Formulas

This problem primarily tests your understanding and application of fundamental properties of determinants, adjoints, and matrix inverses for an n×nn \times n matrix. For an n×nn \times n matrix AA:

  1. Determinant of a scalar multiple of a matrix: det⁡(kA)=kndet⁡(A)\det(kA) = k^n \det(A), where kk is a scalar.
  2. Determinant of an adjoint matrix: det⁡(Adj(A))=(det⁡(A))n−1\det(\text{Adj}(A)) = (\det(A))^{n-1}.
  3. Determinant of an inverse matrix: det⁡(A−1)=1det⁡(A)\det(A^{-1}) = \frac{1}{\det(A)}, provided det⁡(A)≠0\det(A) \neq 0.

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