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.

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