check the template in the picture below.

Please clearer picture thank u
Answer:
I tried to answer but I still would like to know what doesn't make sense.
Considering the identity:


The identity is true whenever both sides are defined. It is not supposed to hold the values when we have division by zero, because it is undefined.
If the value doesn't make sense, it is probably not a function, or it is undefined by the given value of
.
Let's go back to the definition of function again:
A function relates a set of inputs and a set of allowable outputs where each input is related to only one output. It basically defines the relation of ordered pairs.
Hope it helped you a bit.
In order to multiply a matrix by another matrix, we multiply the rows in the first matrix by the columns in the other matrix (How this is done is shown below)
To determine the pairs of matrices that AB=BA, we will determine AB and BA for each of the options below.
For the first option
![A= \left[\begin{array}{cc}1&0&-2&1&\end{array}\right]; B= \left[\begin{array}{cc}5&0&3&2&\end{array}\right] \\](https://tex.z-dn.net/?f=A%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%26-2%261%26%5Cend%7Barray%7D%5Cright%5D%3B%20B%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%260%263%262%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
![AB= \left[\begin{array}{cc}(1\times5)+(0\times3)&(1\times0)+(0\times 2)&(-2\times5)+(1\times3)&(-2\times0)+(1\times2)&\end{array}\right]\\AB= \left[\begin{array}{cc}5+0&0+0&-10+3&0+2&\end{array}\right]\\AB = \left[\begin{array}{cc}5&0&-7&2&\end{array}\right] \\](https://tex.z-dn.net/?f=AB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%281%5Ctimes5%29%2B%280%5Ctimes3%29%26%281%5Ctimes0%29%2B%280%5Ctimes%202%29%26%28-2%5Ctimes5%29%2B%281%5Ctimes3%29%26%28-2%5Ctimes0%29%2B%281%5Ctimes2%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%2B0%260%2B0%26-10%2B3%260%2B2%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%260%26-7%262%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
and
![BA= \left[\begin{array}{cc}(5\times1)+(0\times-2)&(5\times0)+(0\times 1)&(3\times1)+(2\times-2)&(3\times0)+(1\times2)&\end{array}\right]\\BA= \left[\begin{array}{cc}5+0&0+0&3+-4&0+2&\end{array}\right]\\BA = \left[\begin{array}{cc}5&0&-1&2&\end{array}\right] \\](https://tex.z-dn.net/?f=BA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%285%5Ctimes1%29%2B%280%5Ctimes-2%29%26%285%5Ctimes0%29%2B%280%5Ctimes%201%29%26%283%5Ctimes1%29%2B%282%5Ctimes-2%29%26%283%5Ctimes0%29%2B%281%5Ctimes2%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%2B0%260%2B0%263%2B-4%260%2B2%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%260%26-1%262%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
∴ AB≠BA
For the second option
![A= \left[\begin{array}{cc}1&0&-1&2&\end{array}\right]; B= \left[\begin{array}{cc}3&0&6&-3&\end{array}\right] \\](https://tex.z-dn.net/?f=A%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%26-1%262%26%5Cend%7Barray%7D%5Cright%5D%3B%20B%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%260%266%26-3%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
![AB= \left[\begin{array}{cc}(1\times3)+(0\times6)&(1\times0)+(0\times -3)&(-1\times3)+(2\times6)&(-1\times0)+(2\times-3)&\end{array}\right]\\AB= \left[\begin{array}{cc}3+0&0+0&-3+12&0+-6&\end{array}\right]\\AB = \left[\begin{array}{cc}3&0&9&-6&\end{array}\right] \\](https://tex.z-dn.net/?f=AB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%281%5Ctimes3%29%2B%280%5Ctimes6%29%26%281%5Ctimes0%29%2B%280%5Ctimes%20-3%29%26%28-1%5Ctimes3%29%2B%282%5Ctimes6%29%26%28-1%5Ctimes0%29%2B%282%5Ctimes-3%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%2B0%260%2B0%26-3%2B12%260%2B-6%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%260%269%26-6%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
and
![BA= \left[\begin{array}{cc}(3\times1)+(0\times-1)&(3\times0)+(0\times 2)&(6\times1)+(-3\times-1)&(6\times0)+(-3\times2)&\end{array}\right]\\BA= \left[\begin{array}{cc}3+0&0+0&6+3&0+-6&\end{array}\right]\\BA = \left[\begin{array}{cc}3&0&9&-6&\end{array}\right] \\](https://tex.z-dn.net/?f=BA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%283%5Ctimes1%29%2B%280%5Ctimes-1%29%26%283%5Ctimes0%29%2B%280%5Ctimes%202%29%26%286%5Ctimes1%29%2B%28-3%5Ctimes-1%29%26%286%5Ctimes0%29%2B%28-3%5Ctimes2%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%2B0%260%2B0%266%2B3%260%2B-6%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%260%269%26-6%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
Here AB = BA
For the third option
![A= \left[\begin{array}{cc}1&0&-1&2&\end{array}\right]; B= \left[\begin{array}{cc}5&0&3&2&\end{array}\right] \\](https://tex.z-dn.net/?f=A%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%26-1%262%26%5Cend%7Barray%7D%5Cright%5D%3B%20B%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%260%263%262%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
![AB= \left[\begin{array}{cc}(1\times5)+(0\times3)&(1\times0)+(0\times 2)&(-1\times5)+(2\times3)&(-1\times0)+(2\times2)&\end{array}\right]\\AB= \left[\begin{array}{cc}5+0&0+0&-5+6&0+4&\end{array}\right]\\AB = \left[\begin{array}{cc}5&0&1&4&\end{array}\right] \\](https://tex.z-dn.net/?f=AB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%281%5Ctimes5%29%2B%280%5Ctimes3%29%26%281%5Ctimes0%29%2B%280%5Ctimes%202%29%26%28-1%5Ctimes5%29%2B%282%5Ctimes3%29%26%28-1%5Ctimes0%29%2B%282%5Ctimes2%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%2B0%260%2B0%26-5%2B6%260%2B4%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%260%261%264%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
and
![BA= \left[\begin{array}{cc}(5\times1)+(0\times-1)&(5\times0)+(0\times 2)&(3\times1)+(2\times-1)&(3\times0)+(2\times2)&\end{array}\right]\\BA= \left[\begin{array}{cc}5+0&0+0&3+-2&0+4&\end{array}\right]\\BA = \left[\begin{array}{cc}5&0&1&4&\end{array}\right] \\](https://tex.z-dn.net/?f=BA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%285%5Ctimes1%29%2B%280%5Ctimes-1%29%26%285%5Ctimes0%29%2B%280%5Ctimes%202%29%26%283%5Ctimes1%29%2B%282%5Ctimes-1%29%26%283%5Ctimes0%29%2B%282%5Ctimes2%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%2B0%260%2B0%263%2B-2%260%2B4%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%260%261%264%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
Here also, AB=BA
For the fourth option
![A= \left[\begin{array}{cc}1&0&-2&1&\end{array}\right]; B= \left[\begin{array}{cc}3&0&6&-3&\end{array}\right] \\](https://tex.z-dn.net/?f=A%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%26-2%261%26%5Cend%7Barray%7D%5Cright%5D%3B%20B%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%260%266%26-3%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
![AB= \left[\begin{array}{cc}(1\times3)+(0\times6)&(1\times0)+(0\times -3)&(-2\times3)+(1\times6)&(-2\times0)+(1\times-3)&\end{array}\right]\\AB= \left[\begin{array}{cc}3+0&0+0&-6+6&0+-3&\end{array}\right]\\AB = \left[\begin{array}{cc}3&0&0&-3&\end{array}\right] \\](https://tex.z-dn.net/?f=AB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%281%5Ctimes3%29%2B%280%5Ctimes6%29%26%281%5Ctimes0%29%2B%280%5Ctimes%20-3%29%26%28-2%5Ctimes3%29%2B%281%5Ctimes6%29%26%28-2%5Ctimes0%29%2B%281%5Ctimes-3%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%2B0%260%2B0%26-6%2B6%260%2B-3%26%5Cend%7Barray%7D%5Cright%5D%5C%5CAB%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%260%260%26-3%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
and
![BA= \left[\begin{array}{cc}(3\times1)+(0\times-2)&(3\times0)+(0\times 1)&(6\times1)+(-3\times-2)&(6\times0)+(-3\times1)&\end{array}\right]\\BA= \left[\begin{array}{cc}3+0&0+0&6+6&0+-3&\end{array}\right]\\BA = \left[\begin{array}{cc}3&0&12&-3&\end{array}\right] \\](https://tex.z-dn.net/?f=BA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%283%5Ctimes1%29%2B%280%5Ctimes-2%29%26%283%5Ctimes0%29%2B%280%5Ctimes%201%29%26%286%5Ctimes1%29%2B%28-3%5Ctimes-2%29%26%286%5Ctimes0%29%2B%28-3%5Ctimes1%29%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%2B0%260%2B0%266%2B6%260%2B-3%26%5Cend%7Barray%7D%5Cright%5D%5C%5CBA%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%260%2612%26-3%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
Here, AB≠BA
Hence, it is only in the second and third options that AB = BA
and ![A= \left[\begin{array}{cc}1&0&-1&2&\end{array}\right]B= \left[\begin{array}{cc}5&0&3&2&\end{array}\right] \\](https://tex.z-dn.net/?f=A%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%26-1%262%26%5Cend%7Barray%7D%5Cright%5DB%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D5%260%263%262%26%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
Learn more on matrices multiplication here: brainly.com/question/12755004