CSIR NET July 2026 Memory based Question | Part 4

[00:00:00]Hello students, welcome to the part four of the memory based questions of the CSNet July 2026. Myself Dr. Harishkar.

[00:00:09]You can follow and subscribe my YouTube channel. The part one, part two and the part three you can also find at my YouTube channel Dr. Harish G. Today I will explain you how you can solve the questions related to the linear algebra.

[00:00:25]The first question is related to the rank of the matrix and the second is related to the minimal polinomial. How you can solve both the question in just the 15 second time period. It's a very very simple. Now read the statement. V is the vector space of the polomial less than and equal to 31. What does it implies? That means the dimension of the v is my 31 and t is my non0ero t² is equal to zero.

[00:00:56]The maximum rank of the very simple t power something is my zero who what first things come in your mind related to the matrix. Any idea what first things come in your mind related to the matrix always be the nil potent fine matrix t power k will be zero it look like a n potent now look at that how you can solve the question in a 15cond time period if t power k will be zero and dimension of the vector space v is my n then we can find the minimum minimum value of the rank and the maximum value of the rank both in a single line.

[00:01:45]Minimum is always one maximum will be k -1 n / k. Fine. Now if you compare them, what is my k? K is my 2. What is the n?

[00:01:59]n is my 31. Fine.

[00:02:02]Maximum you can substitute in the right hand side. So k - 1 31 divided by 2 to 15.5 but we need a maximum that will be less than or equal to 15. So B is the right answer of the problem.

[00:02:22]Scholars same method sol let me know in the comment box.

[00:02:38]Fine. Now look at the second one. For N cross N real matrix mu is the minimal polomial. Which of the following statement is or are correct? Again, it's a 15-second time period. You can solve the problem. But make sure students all these questions are the memory based questions.

[00:02:57]If you feel that any of the statement is not a correct, send the correct statement in the comment box. I will upload the next video in a 15 minutes time periods for all of you. Tell you minimal polinomial is given to you. If x first option if x square + 1 is a minimal polomial what does it means?

[00:03:18]That means a square + i must be zero. If a square + i must be zero then what is the value of the a square? It's a minus of i and your target is to find the value of a4. Clearly say a4 must be equal to identity. Yes, it's the correct state. Look at the second option. Again x² + a is my minimal polomial. So I can say a square must be equal to minus of a. Now your target is to write the power 8. I can take the power four on the both side. It's a a4. How you can find the a4? I can squaring on the both side.

[00:04:00]That will be a square and a square will be my minus of a which is not always equal to identity. Fine. So that means the second option is also cancel clear because values values are 0 and 1. So you can consider any matrix like of this case. Is it okay? Now third option x² - x. So again you can see a square - a + i will be zero. You can find the value of the a square. It's a I minus a your target is to power six. So I can cubing on the both side but it will take some time.

[00:04:49]Now I will tell you the another way minimal polides theating polomial. Now can you think any polomial which involvement of the x² - x + 1? Any polinomial which involvement of the x square - 1 + 1 as a factor. Yes.

[00:05:13]Aqu + b c that will be a + b into a square minus a b + b. So what does it means? If x² + - x + 1 is a minimal polomial that means a cube + i must be zero. So what does it implies? That implies a cube will be minus of i. And if you scaling on the both side you will get the correct answer.

[00:05:41]Look at the last option again. Do you get any polomial which involvement of the x² + x + 1? Yes, you can easily do that is a x cub minus 1.

[00:05:56]Fine. Now, because x square + x + 1 is a minimal polinomial, xq - 1 is analyating polinomial. Therefore, aquus 1 will be zero. That implies a cube will be my identity. Now you also need a cube as my identity. Right answers are a, c and d are my correct answers. So let me know if you have some more questions send me in the comment box or comments. You can send me the comment box. I will answer yours very soon. I hope you can subscribe my YouTube channels and you can mark interested to watch more videos in the comment box.

[00:06:38]Best of luck students. Happy.