Let M be a 2   × 2 symmetric matrix with integer entries. Then M is invertible if

Let M be a 2 ×2 symmetric matrix with integer entries. Then M is invertible if
  1. The first column of M is the transpose of the second row of M
  2. The second row of M is the transpose of the first column of M
  3. M is a diagonal matrix with non - zero entries in the main diagonal
  4. The product of entries in the main diagonal of M is not the square of an integer

Solution

Let M=accb (where a, b, c ∈I )
Then Det M=ab-c2
If a=b=c, DetM=0
If c=0, a, b ≠0, Det M≠0
If ab ≠  square of integer, Det M≠0

Asked in: JEE Advanced 2014 (Paper 1)

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