[Math] Find a general 2×2 matrix where A = adj(A)

Question

I know how to find the adjoint of $2\times 2$ matrix but I'm at a loss for finding a general $2\times 2$ matrix where $A = adj(A)$.

Thanks for your help!

Answer 1

If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ then $\operatorname{adj} A = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. So you need to find $a,b,c,d$ such that $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. $$ In other words, you have to solve the system of linear equations $$ \begin{align*} a &= d, \\ b &= -b, \\ c &= -c, \\ d &= a. \end{align*} $$ You take it from here.