Question

Let M be a $2\times 2$ 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 column of M is the transpose of the first row of M.
  
• C. M is a diagonal matrix with nonzero 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 nonzero entries in the main diagonal.

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

$M=\left[\begin{array}{cc}a& k\\ k& b\end{array}\right]$
(A) $\left[\begin{array}{c}a\\ k\end{array}\right]=\left[\begin{array}{c}k\\ b\end{array}\right]\Rightarrow a=k=b\Rightarrow \left|M\right|=0$
(B) $\left[\begin{array}{c}k\\ b\end{array}\right]=\left[\begin{array}{c}a\\ k\end{array}\right]\Rightarrow k=a=b\Rightarrow \left|M\right|=0$
(C) $M=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]\Rightarrow \left|M\right|\ne 0$
(D) $M=\left[\begin{array}{cc}a& k\\ k& b\end{array}\right]\Rightarrow \left|M\right|=ab-k^2\ne 0$