# no. of matrices

#### jacks

##### Well-known member
How can i find all matrix $A$ of order $2\times 2$ that satisfy the $A^2 = \begin{pmatrix}1 & 1\\ 0 & 1 \end{pmatrix}$

#### Sudharaka

##### Well-known member
MHB Math Helper
How can i find all matrix $A$ of order $2\times 2$ that satisfy the $A^2 = \begin{pmatrix}1 & 1\\ 0 & 1 \end{pmatrix}$
Hi jacks,

Let $$A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$$. Then $$A^2 = \begin{pmatrix}a^2+bc & ab+bd\\ ac+cd & bc+d^2\end{pmatrix}$$

Since, $$A^2 = \begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$$

$a^2+bc=1~~~~~~~(1)$

$ab+bd=1~~~~~~~(2)$

$bc+d^2=1~~~~~~~~(3)$

$ac+cd=0~~~~~~~~(4)$

From (1) and (3), $$a=\pm d$$. But (4) implies that, $$a=d$$. Then substituting for $$d$$ in (4), $$ac=0$$. If $$a=0\Rightarrow d=0$$ and we get a contradiction from (2). Therefore, $$c=0$$. Then by (1) and (3), $$a=d=\pm 1$$. Finally by (2), $$b=\pm\frac{1}{2}$$. So we have two sets of answers,

$a=d=1,~b=\frac{1}{2},~c=0\mbox{ or }a=d=-1,~b=-\frac{1}{2},~c=0$