Q92 / 180 · KCET 2021 · Mathematics

Let $M$ be $2 \times 2$ symmetric matrix with integer entries, then $M$ is invertible if

  • A
    the first column of $M$ is the transpose of second row of $M$.
  • B
    the second row of $M$ is the transpose of first column of $M$.
  • Correct answer
    C
    $M$ is diagonal matrix with non-zero entries in the principal diagonal.
  • D
    The product of entries in the principal diagonal of $M$ is the product of entries in the other diagonal.

Solution

Let a symmetric matrix $M=\left[\begin{array}{ll} a & c \\ c & b \end{array}\right]$ For matrix to be invertible, determinant must not be equal to zero. $\begin{aligned} & &|M| &=a b-c^{2} \neq 0 \\ \Rightarrow & & a b & \neq c^{2} \end{aligned}$ Therefore, $M$ is a diagonal matrix with non-zero entries in the main diagonal and the product of entries in the main diagonal of $M$ is not the square of an integer.

Hence, the correct answer is option: 'C'.

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