<s>[INST] Square root of Matrix $A=\begin{bmatrix} 1 &2 \\ 3&4 \end{bmatrix}$ Find Square root of Matrix $A=\begin{bmatrix} 1 &2 \\   3&4  \end{bmatrix}$ without using concept of Eigen Values and Eigen Vectors
I assumed its square root as $$B=\begin{bmatrix} a &b \\   c&d  \end{bmatrix}$$
hence
$$B^2=A$$ $\implies$
$$\begin{bmatrix} a^2+bc &b(a+d) \\   c(a+d)&d^2+bc  \end{bmatrix}=\begin{bmatrix} 1 &2 \\   3&4  \end{bmatrix}$$
So
$$a^2+bc=1 \tag{1}$$
$$b(a+d)=2 \tag{2}$$
$$c(a+d)=3 \tag{3}$$
$$d^2+bc=4 \tag{4}$$  From $(2)$ and $(3)$ we get
$$\frac{b}{c}=\frac{2}{3}$$
Let $b=2k $ and $c=3k$ Then
$$a^2=1-6k^2$$ and $$d^2=4-6k^2$$  So
$$B=\begin{bmatrix} \sqrt{1-6k^2} &2k\\   3k&\sqrt{4-6k^2}  \end{bmatrix}$$ 
So
$$B^2=\begin{bmatrix} 1 &2k \left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right) \\   3k \left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right)&4  \end{bmatrix}=\begin{bmatrix} 1 &2 \\   3&4  \end{bmatrix}$$
Now we have to solve for $k$ using equation
$$ \left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right)=\frac{1}{k} \tag{5}$$ Also
$$\left(\sqrt{1-6k^2}+\sqrt{4-6k^2}\right) \times \left(\sqrt{4-6k^2}-\sqrt{1-6k^2}\right)=3$$ So from $(5)$
$$\left(\sqrt{4-6k^2}-\sqrt{1-6k^2}\right)=3k \tag{6}$$
Subtracting $(6)$ from $(5)$ we get
$$2\sqrt{1-6k^2}=\frac{1}{k}-3k$$  Squaring Both sides we get
$$33k^4-10k^2+1=0$$ we get
$$k^2=\frac{5 \pm i \sqrt{8}}{33}$$ From this we get $k$. is there any other simpler way to find ?
 [/INST] @ Umesh shankar, this is exactly what should not be done (if you want to make progress)!!
Note that $A^2=trace(A)A-\det(A)I=5A+2I$. Since $B^2=A$ we deduce that $A,B$ commute and since $A$ is not a scalar matrix, $B$ is in the form $B=aI+bA$. Then $(aI+BA)^2=A$, that is $a^2I+b^2(5A+2I)+2abA=A$, that implies
$$a^2+2b^2=0\;,\;5b^2+2ab=1.$$
We deduce $33b^4-10b^2+1=0$,....... and, finally, the four solutions.
EDIT. 
Recall that the determinant and the trace of a matrix do not change after a change of basis.
Proposition 1. If $A\in M_2$ is not a scalar matrix, then $I,A$ is a basis of the vector space $E=\{X|AX=XA\}$.
Proof. With a change of basis, we may assume that $A=\begin{pmatrix}0&a\\1&b\end{pmatrix}$ (there is $u$ s.t. $u,Au$ is a basis). Let $X=\begin{pmatrix}x&y\\z&t\end{pmatrix}$. Then $AX=XA$ is equivalent to $y=az,x+bz=t$, that is $2$ linear independent linear relations; thus $dim(E)=4-2=2$.
Proposition 2. If $A\in M_2$ is not a scalar matrix, then $A^2=trace(A)A-\det(A)I$.
Proof. As above, we may assume that $A=\begin{pmatrix}0&a\\1&b\end{pmatrix}$.
$A^2=\begin{pmatrix}a&ab\\b&a+b^2\end{pmatrix}=bA+aI=trace(A)A-\det(A)I$.
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