Question

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 the second row of M
• B. the second row of M is the transpose of the first column of M
• C. M is a diagonal matrix with non - zero entries in the main diagonal
• D. the product of entries in the main diagonal of M is not the square of an integer

Answer 1

Answer: (C), (D)

The correct options are
C

M is a diagonal matrix with non - zero entries in the main diagonal

D

the product of entries in the main diagonal of M is not the square of an integer

Let $M=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$ (where a,b,c $\in$ I)
(a) If the first column of M is the transpose of the second row of M, then $\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}b\\ c\end{array}\right]$
$\therefore a=b=c$
Thus, $det.\left(M\right)=ac-b^2=0$
Hence, M is not invertible.
(b) If the second row of M is the transpose of the first column of M, then $\left[\begin{array}{cc}b& c\end{array}\right]=\left[\begin{array}{cc}a& b\end{array}\right]$
$\therefore a=b=c$
Thus, $det.\left(M\right)=ac-b^2=0$
Hence, M is not invertible.
(c) If $M=\left[\begin{array}{cc}a& 0\\ 0& c\end{array}\right]$, with a,c $\ne$ 0, then
Thus, $det.\left(M\right)=ac\ne 0$
Hence, M is invertible.
(d) If product of elements in main diagonal which (ac) is not perfect square, then
$det.\left(M\right)=ac-b^2\ne 0$
Hence, M is invertible.