Question

Let n be a 2 x 2 complex matrix such that n2 = 0. Prove that either n = 0 or n is similar over c to [0 01

Answer 1

n = 0 if n be a 2 x 2 complex matrix such that n² = 0.

Step-by-step explanation:

n is  2 * 2

Let say  n  =    $\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$

=> n = ad - bc

n² =   $\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \times \left[\begin{array}{cc}a&b\\c&d\end{array}\right]$

= $\left[\begin{array}{cc}a^2 + bc&ab+bd\\ca + cd&bc + d^2\end{array}\right]$

n² = 0

=> (a² + bc)(bc + d²) - (ca + cd)(ab + bd) = 0

=> a²d²  + bc(a² + d²) + b²c²  -  bc(a + d)² = 0

=>  a²d²  + bc(a² + d²) + b²c²  - bc(a² + d² + 2ad) = 0

=>  a²d² +  b²c² - 2abcd = 0

=> (ad - bc)² =0

=> ad - bc = 0

=> n = 0

QED

proved

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