Question

Consider the matrix equation X² = I, I being a 2x2 unit matrix, ‘X’ is a real matrix (all the elements being real). Which of the following is/are correct?

• A. The equation has exactly two solutions
• B. The equation has infinitely many Solutions
• C. $x=\left[\begin{array}{cc}\sqrt{2}-\sqrt{3}& 4\sqrt{2}\\ \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}& \sqrt{3}-\sqrt{2}\end{array}\right]$
• D. $x=\left[\begin{array}{cc}\sqrt{3}-\sqrt{2}& \frac{3}{2}-1\\ 4& \sqrt{2}-\sqrt{3}\end{array}\right]$

Answer 1

Answer: (B), (C), (D)

The correct options are
B The equation has infinitely many Solutions
C $x=\left[\begin{array}{cc}\sqrt{2}-\sqrt{3}& 4\sqrt{2}\\ \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}& \sqrt{3}-\sqrt{2}\end{array}\right]$
D $x=\left[\begin{array}{cc}\sqrt{3}-\sqrt{2}& \frac{3}{2}-1\\ 4& \sqrt{2}-\sqrt{3}\end{array}\right]$

c, d can be obtained by angle verification
$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ then $x^2=I$ given
$a^2+bc=1=bc+d^2\Rightarrow a=\pm d$
b(a +d) = c(a + d)=0 given b = c = 0 or a +d = 0
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}a& b\\ c& -a\end{array}\right](a^2+b=1)$
$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$ are solution