Question

Let M be the set of all possible $2\times 2$ matrix A of integer entries such that $A{A}^T=I$ where $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ then

• A. Number of matrices in set M is 8.
• B. Number of matrices in set M is 4
• C. Number of matrices in set M such that $|A-I|\ne 0$ is 3
• D. Number of matrices in set M such that $|A-I|\ne 0$ is 2

Answer 1

Answer: (A), (C)

The correct options are
A

Number of matrices in set M is 8.

C

Number of matrices in set M such that $|A-I|\ne 0$ is 3

$LetA=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]Then\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\therefore a^2+b^2=1ac+bd=0c^2+d^2=1det(A-I)=\left|\begin{array}{cc}a-1& b\\ c& d-1\end{array}\right|=ad-a-d-bc+1\begin{array}{ccccc}a& b& c& d& det(A-I)\\ 1& 0& 0& 1& 0\\ 1& 0& 0& -1& 0\\ -1& 0& 0& 1& 0\\ -1& 0& 0& -1& 4\\ 0& 1& 1& 0& 0\\ 0& 1& -1& 0& 2\\ 0& -1& 1& 0& 2\\ 0& -1& -1& 0& 0\end{array}$