This question is about Eigen values and Eigen vectors here we want to find the Eigen values of this matrix and for each again, really find a basis for the ANC. In space. Recall that to find the Eigen values, we have to solve the characteristic equation which is characteristic polynomial equal zero. What is the characteristic polynomial? Again? It is in this case a cubic polynomial of the scalar lambda. Mhm. Defined by the determinant of a man assigned to. I worked as the identity nature. Yeah, this determinant is the determinant of this matrix here, which is a menace lambda I or just hey with negative land was added onto the middle or the deck. And or entry. Now I'm going to use the co factor expansion definition of the determinant define the three by three determinant here, one minus lambda. I'm going to choose this column. Take the co factor expansion of the column, which only has a non zero only has one non zero entry. The one corresponding to one climbed up and that will be one minus lambda times the co factor of what might have slammed down which is a positive one times the determinant of the sub matrix we get by deleting the column and row of this entry. So that's the co factor expansion of the middle room. Middle column one. Medicine. Bad times. It's co factor. Now we have to evaluate this two by two determinant, which is just a demon sbc. So here I've expanded the foreman assigned attempts. One man is named after and once we factor that completely, we end up with Yeah, our three organ values. One, two and right. So our matrix A. Has these three Eigen values 12 and three. Now we want to find Eigen vectors. So an Eigen vector corresponding to the first dragon volume is a solution of this system. The system is happened homogeneous. So I must have at least one solution, which is the zero solution trivial solution in other words. But since we said that since the lambda equals one was a solution to the characteristic equation. A menace, I has a determinant of zero and thus must be uninhabitable. And so there is not just one solution in this case, but infinitely many. And in fact we go about the process of finding Eigen values like this specifically so that we would have infinitely many solutions and not just the zero vector. We want non zero vectors, non zero vector solutions to this. And the only way to have that is this where Mhm. By the way, Eigen vectors already find that's being non zero. So how do we find the solutions? Well, one way is to will reduce those matrix to roll reduced echelon form. This is my real reduced echelon form which uh well from here to here, I haven't written down my steps but all I did was subtract this row times three from this room and then switch the rose. So this tells me not. Ah the no space of this matrix is the same as the null space of this matrix. And so the solution I want here is the same from the solution set of this equation is the same as the solution set of this equation. And so uh if I parameter rise my solutions I would find that all the solutions can be written as a scalar multiple of this factor 010 X. And Z must necessarily be zero. And why is my pretty variable? I have just 13 variable. There's only going to be one vector in the uh basis of the organ space because all vectors can be recognized This killer multiple of just one. A linear combination of one victory. Now let's move on to the second african space. The one corresponding to the second item value. This again space is the set of all vectors that solve this system. What is that now? We have the second organ Valium. Thanks to find all these this infinite number of solutions, we will reduce this matrix that I like to call the Eigen matrix of this Eigen value. Since we already call we have already all these times for with agonize the prefix why not call this an Eigen matrix Now I will reduce that to find a solution of the homogeneous system corresponding to this matrix. After five or six rural reductions. I would get this rail reduced echelon form. It was solution set is the set of all scalar multiples hug this vector. And so this vector is a basis factor for the dragon space. It forms a set all on its own. That is a basis. The second idea. No, The last. The Eigen matrix is a minus three I. Which is this. So a possible solution vector is minus 111 and again we have one free variable. And so the basis has only one vector, which is an inspector. There is our third basis. Yeah. And as a fun fact, these three vectors could be used to diagonal. Is this matrix, but that's a topic for another video.