# Matrices

#### yo79

Find two 2x2 matrices with real entries A, B such that $$\displaystyle A^2+B^2=\left( \begin{array}{cc}2&3\\3&2\end{array}\right)$$. (I need just an example.)

#### agentredlum

Math Team
yo79 said:
Find two 2x2 matrices with real entries A, B such that $$\displaystyle A^2+B^2=\left( \begin{array}{cc}2&3\\3&2\end{array}\right)$$. (I need just an example.)
Nice question!

Well i played around with it for a while till i realized we could work with rows as one way to get an answer.

$$\displaystyle $\sqrt{2} \ \ \frac{3}{\sqrt{2}} \\ \\ \ 0 \ \ \ \ 0$ \cdot $\sqrt{2} \ \ \frac{3}{\sqrt{2}} \\ \\ \ 0 \ \ \ \ 0$ \ + \ $0 \ \ \ \ \ 0 \\ \\ \frac{3}{\sqrt{2}} \ \ \sqrt{2}$ \cdot $0 \ \ \ \ \ 0 \\ \\ \frac{3}{\sqrt{2}} \ \ \sqrt{2}$ \ = \ $2 \ \ 3 \\ 3 \ \ 2$$$ #### mathbalarka

Math Team
Nice question with a nice solution!

Thank you!

#### agentredlum

Math Team
You're welcome. Just in case you haven't noticed , we now have an easy solution to an infinite amount of problems of this form.

For $$\displaystyle \ p \ \ne \ 0$$

IF

$$\displaystyle A^2 \ + \ B^2 \ = \ $p \ q \\ q \ p$$$

THEN a solution will be

$$\displaystyle A \ = $\sqrt{p} \ \ \frac{q}{ \sqrt{p}} \\ \ 0 \ \ \ \ 0$$$

$$\displaystyle B \ = \ $\ \ 0 \ \ \ \ 0 \\ \frac{q}{ \sqrt{p}} \ \ \sqrt{p}$$$

We can also transpose both A , B and get another family of solutions. Then we would be working with columns.

An interesting question to ask is 'what if $$\displaystyle \ p \ = \ 0 \ \ \text{AND} \ \ q \ \ne \ 0 \$$?'

A place to start could be ,

Find a 2×2 matrix A such that

$$\displaystyle A^2 \ = \ $0 \ 1 \\ 1 \ 0$$$

Can it be found ? #### yo79

Of course not, if A has real entries ( det(A)^2=-1)!

#### CRGreathouse

Forum Staff
agentredlum said:
Find a 2×2 matrix A such that

$$\displaystyle A^2 \ = \ $0 \ 1 \\ 1 \ 0$$$

Can it be found ?
Not in the real numbers. There is a unique complex solution up to the order of the variables.

#### agentredlum

Math Team
Can we find A with complex entries? Math Team

#### agentredlum

Math Team
Find 4 distinct matrices A such that

$$\displaystyle A^2 \ = \ $4 \ 0 \\ 0 \ 4$$$ 