# Matrices which are their own inverses

• February 19th 2009, 06:26 PM
Snooks02
Matrices which are their own inverses
I need to find all 2x2 matrices A,
(a b)
(c d)
such that A^2=I2, the 2x2 identity matrix.
• February 19th 2009, 09:15 PM
tah
this means that A is equal to its inverse. So you may write the expression of $A^{-1}$ and deduce 4 equations from the identification of it with A. It remains to solve these equations
• February 19th 2009, 09:57 PM
Jhevon
Quote:

Originally Posted by Snooks02
I need to find all 2x2 matrices A,
(a b)
(c d)
such that A^2=I2, the 2x2 identity matrix.

Just to flesh out what tah said:

you have $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

now, compute the matrix product on the left hand side, then equate corresponding components to get your 4 equations that tah spoke of. solve this system for your solution
• February 19th 2009, 10:51 PM
math2009
$A=\begin{bmatrix} a&b \\ \frac{1-a^2}{b} &-a \end{bmatrix}\ or\ \pm I_2$