Search any question & find its solution
Question:
Answered & Verified by Expert
Let $M$ be $2 \times 2$ symmetric matrix with integer entries, then $M$ is invertible if
Options:
Solution:
2534 Upvotes
Verified Answer
The correct answer is:
$M$ is diagonal matrix with non-zero entries in the principal diagonal.
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.
$M=\left[\begin{array}{ll}
a & c \\
c & b
\end{array}\right]$
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.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.