M = \begin{bmatrix}
2 & 2
1 & 1
\end{bmatrix}

Solve the equation: X^5=A, X \epsilon M_2 (R)
__________________________________________________ ________

If X = \begin{bmatrix}
a & b
c & d
\end{bmatrix}

det(A)=0
det(X^5)=[det(X)]^5=0, so det(X)=0

Using Hamilton Cayley theorem:
X^2 = (a d)X det(X) = (a d)X
X^5 = X^4 \times X = (X^2)^2 \times X = (a d)^2 X^2 \times X = (a d)^2 (a d) X^2 = (a d)^3 X^2 = (a d)^4 X

How should I proceed next?

X^5 = (a+d)^4 X

I don't know what went wrong while editing...