MHB Finding All $2\times 2$ Matrices That Satisfy $A^2=A$

[juantheron]

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]

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,

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