Answer:
Options 1 and 3.
Step-by-step explanation:
By definition the product between two matrices is:
Let's suppose both matrices are 2x2,
![A=\left[\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Da_%7B11%7D%26a_%7B12%7D%5C%5Ca_%7B21%7D%26a_%7B22%7D%5Cend%7Barray%7D%5Cright%5D)
![B=\left[\begin{array}{cc}b_{11}&b_{12}\\b_{21}&b_{22}\end{array}\right]](https://tex.z-dn.net/?f=B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Db_%7B11%7D%26b_%7B12%7D%5C%5Cb_%7B21%7D%26b_%7B22%7D%5Cend%7Barray%7D%5Cright%5D)
The product between A and B is:
![AB=\left[\begin{array}{cc}a_{11}b_{11}+a_{12}b_{21}&a_{11}b_{12}+a_{12}b_{22}\\a_{21}b_{11}+a_{22}b_{21}&a_{21}b_{12}+a_{22}b_{22}\end{array}\right]](https://tex.z-dn.net/?f=AB%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7Da_%7B11%7Db_%7B11%7D%2Ba_%7B12%7Db_%7B21%7D%26a_%7B11%7Db_%7B12%7D%2Ba_%7B12%7Db_%7B22%7D%5C%5Ca_%7B21%7Db_%7B11%7D%2Ba_%7B22%7Db_%7B21%7D%26a_%7B21%7Db_%7B12%7D%2Ba_%7B22%7Db_%7B22%7D%5Cend%7Barray%7D%5Cright%5D)
IMPORTANT: It's not the same AB then BA the results of both products are differents.
Now we are going to analyze every option:

Option 1:
![A=\left[\begin{array}{cc}5&5\\-4&-4\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%265%5C%5C-4%26-4%5Cend%7Barray%7D%5Cright%5D)
![A.A=\left[\begin{array}{cc}5&5\\-4&-4\end{array}\right] .\left[\begin{array}{cc}5&5\\-4&-4\end{array}\right] =\\\\=\left[\begin{array}{cc}5.5+5(-4)&5.5+5(-4)\\(-4).5+(-4).(-4)&(-4).5+(-4)(-4)\end{array}\right] \\\\=\left[\begin{array}{cc}25-20&25-20\\-20+16&-20+16\end{array}\right] \\\\=\left[\begin{array}{cc}5&5\\-4&-4\end{array}\right]=A](https://tex.z-dn.net/?f=A.A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%265%5C%5C-4%26-4%5Cend%7Barray%7D%5Cright%5D%20.%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%265%5C%5C-4%26-4%5Cend%7Barray%7D%5Cright%5D%20%3D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5.5%2B5%28-4%29%265.5%2B5%28-4%29%5C%5C%28-4%29.5%2B%28-4%29.%28-4%29%26%28-4%29.5%2B%28-4%29%28-4%29%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D25-20%2625-20%5C%5C-20%2B16%26-20%2B16%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%265%5C%5C-4%26-4%5Cend%7Barray%7D%5Cright%5D%3DA)
We can see that A.A=A then this is the correct option.
Option 2:
![A=\left[\begin{array}{cc}6&5\\5&6\end{array}\right]\\\\A.A=\left[\begin{array}{cc}6&5\\5&6\end{array}\right].\left[\begin{array}{cc}6&5\\5&6\end{array}\right]\\\\=\left[\begin{array}{cc}6.6+5.5&6.5+5.6\\5.6+6.5&5.5+6.6\end{array}\right]\\\\=\left[\begin{array}{cc}36+25&30+30\\30+30&25+36\end{array}\right]\\\\=\left[\begin{array}{cc}61&60\\60&61\end{array}\right]\neq A](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6%265%5C%5C5%266%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5CA.A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6%265%5C%5C5%266%5Cend%7Barray%7D%5Cright%5D.%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6%265%5C%5C5%266%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6.6%2B5.5%266.5%2B5.6%5C%5C5.6%2B6.5%265.5%2B6.6%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D36%2B25%2630%2B30%5C%5C30%2B30%2625%2B36%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D61%2660%5C%5C60%2661%5Cend%7Barray%7D%5Cright%5D%5Cneq%20A)
Then this option is incorrect.
Option 3:
![A=\left[\begin{array}{cc}0,5&-0,5\\-0,5&0,5\end{array}\right] \\\\A.A=\left[\begin{array}{cc}0,5&-0,5\\-0,5&0,5\end{array}\right].\left[\begin{array}{cc}0,5&-0,5\\-0,5&0,5\end{array}\right]\\\\=\left[\begin{array}{cc}0,5.0,5+(-0,5).(-0,5)&0,5.(-0,5)+(-0,5).(0,5)\\(-0,5).0,5+0,5.(-0,5)&(-0,5).(-0,5)+0,5.0,5\end{array}\right]\\\\=\left[\begin{array}{cc}0,25+0,25&-0,25-0,25\\-0,25-0,25&0,25+0,25\end{array}\right]\\\\=\left[\begin{array}{cc}0,5&-0,5\\-0,5&0,5\end{array}\right]= A](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5%26-0%2C5%5C%5C-0%2C5%260%2C5%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5CA.A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5%26-0%2C5%5C%5C-0%2C5%260%2C5%5Cend%7Barray%7D%5Cright%5D.%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5%26-0%2C5%5C%5C-0%2C5%260%2C5%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5.0%2C5%2B%28-0%2C5%29.%28-0%2C5%29%260%2C5.%28-0%2C5%29%2B%28-0%2C5%29.%280%2C5%29%5C%5C%28-0%2C5%29.0%2C5%2B0%2C5.%28-0%2C5%29%26%28-0%2C5%29.%28-0%2C5%29%2B0%2C5.0%2C5%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C25%2B0%2C25%26-0%2C25-0%2C25%5C%5C-0%2C25-0%2C25%260%2C25%2B0%2C25%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5%26-0%2C5%5C%5C-0%2C5%260%2C5%5Cend%7Barray%7D%5Cright%5D%3D%20A)
We can see that this option is also correct.
Option 4:
![A=\left[\begin{array}{cc}0,5&0,5\\-0,5&0,5\end{array}\right] \\\\A.A=\left[\begin{array}{cc}0,5&0,5\\-0,5&0,5\end{array}\right].\left[\begin{array}{cc}0,5&0,5\\-0,5&0,5\end{array}\right]\\\\=\left[\begin{array}{cc}0,5.0,5+0,5.(-0,5)&0,5.0,5+0,5.0,5\\(-0,5).0,5+0,5.(-0,5)&(-0,5).0,5+0,5.0,5\end{array}\right]\\\\=\left[\begin{array}{cc}0,25-0,25&0,25+0,25\\-0,25-0,25&-0,25+0,25\end{array}\right]\\\\=\left[\begin{array}{cc}0&0,5\\-0,5&0\end{array}\right]\neq A](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5%260%2C5%5C%5C-0%2C5%260%2C5%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5CA.A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5%260%2C5%5C%5C-0%2C5%260%2C5%5Cend%7Barray%7D%5Cright%5D.%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5%260%2C5%5C%5C-0%2C5%260%2C5%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C5.0%2C5%2B0%2C5.%28-0%2C5%29%260%2C5.0%2C5%2B0%2C5.0%2C5%5C%5C%28-0%2C5%29.0%2C5%2B0%2C5.%28-0%2C5%29%26%28-0%2C5%29.0%2C5%2B0%2C5.0%2C5%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%2C25-0%2C25%260%2C25%2B0%2C25%5C%5C-0%2C25-0%2C25%26-0%2C25%2B0%2C25%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0%260%2C5%5C%5C-0%2C5%260%5Cend%7Barray%7D%5Cright%5D%5Cneq%20A)
Then this option is incorrect.
Option 5:
![A=\left[\begin{array}{cc}-6&-6\\5&5\end{array}\right]\\\\A.A=\left[\begin{array}{cc}-6&-6\\5&5\end{array}\right].\left[\begin{array}{cc}-6&-6\\5&5\end{array}\right]\\\\=\left[\begin{array}{cc}(-6).(-6)+(-6).5&(-6).(-6)+(-6).5\\5.(-6)+5.5&5.(-6)+5.5\end{array}\right]\\\\=\left[\begin{array}{cc}36-30&36-30\\-30+25&-30+25\end{array}\right]\\\\=\left[\begin{array}{cc}6&6\\-5&-5\end{array}\right]\neq A](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-6%26-6%5C%5C5%265%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5CA.A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-6%26-6%5C%5C5%265%5Cend%7Barray%7D%5Cright%5D.%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D-6%26-6%5C%5C5%265%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%28-6%29.%28-6%29%2B%28-6%29.5%26%28-6%29.%28-6%29%2B%28-6%29.5%5C%5C5.%28-6%29%2B5.5%265.%28-6%29%2B5.5%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D36-30%2636-30%5C%5C-30%2B25%26-30%2B25%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6%266%5C%5C-5%26-5%5Cend%7Barray%7D%5Cright%5D%5Cneq%20A)
Then this option is incorrect.