# Thread: Matrices which are their own inverses

1. ## 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.

2. 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

3. 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

4. $A=\begin{bmatrix} a&b \\ \frac{1-a^2}{b} &-a \end{bmatrix}\ or\ \pm I_2$

### is it possible for a matrix to be its own inverse

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