Find Eigenvalues of 3×3 Matrix MUCH Faster ⚡

追加: 2026-05-26

[00:00:00]Hi students. So in this next short video, next important area from the linear algebra that is the igon values of a 3 +3 matrix. Yes, normally when it comes to igon value problem, we can have many properties as well to solve the igon values given MCQ or MSQ questions or NAT question. But at the same times really sometimes we need the characteristic equation. Now normally when we talk about characteristic equation for any matrix, it is determinant of a minus lambda equal to zero. And what I have observed is whether it is 2 +2 or 3 + 3 or any order matrix students usually take up through this characteristic equation and forming the characteristic equation every time by determinant a minus lambda would be timeconuming task for 3 + 3. What is the direct characteristic equation? You can always note down lambda q minus beta 1 lambda square plus beta 2 lambda minus beta 3 equal to 0 and form the characteristic equation directly. What are these betas? Beta 1 is the trace of the matrix. Okay. So now let's continue the same example. Well, if I wanted a trace of this matrix sum of diagonal numbers, this becomes how much? Two.

[00:00:54]Okay. 1 + 1 + 0. 1 + 1 + 0. Okay. Now, the next is beta 2. That is a sum of the minus of diagonals element. Okay. So, when you talk about the diagonal elements, what are going to be the minor? This is the first diagonal element. Minor means deleting first row and first column. What I'm going to get?

[00:01:09]Cross multiply. I get 0 minus one. Okay.

[00:01:11]Cross multiply. Or if you want it for the first time, I will write down this minor. Okay. Calculations you can do.

[00:01:17]Plus, what is the second diagonal number? This is second diagonal number.

[00:01:19]Deleting second row second column what do you get? Okay. 1 - 2 1 0 1 and -2 1 and 0 here. Plus if I come to the next diagonal element, the third diagonal element is 0. Deleting the third row, third column, what do you get? 1 1 0 1 that I'm going to place up here. Okay.

[00:01:35]The calculation says that this is minus one. Okay. Next one I'm getting - + 2 and next one I should be getting 1 - 0 which is 1. So this turns out to be how much? This turns out to be two. Okay.

[00:01:46]Beta 3 is a determinant. So at least trace and determinant is easy. Only a bit of calculation. Sum of minus you have to calculate for all diagonal elements. Okay. Now determinant you can all calculate it's a easy calculation and you are going to get a determinant of this matrix. I'll leave it to you that is going to be zero. Now I'll straightforward put down these numbers in this characteristic equation. Okay.

[00:02:03]Now you also might be thinking sir how cubic equation we can solve. Okay don't worry if there's a cubic equation it will be easily factorizable. Why?

[00:02:09]Because if I put beta 3 equal to 0 only this part will be left and out of this part when I take lambda common I'm going to get a quadratic factor. lambda square - 2 lambda + 2 equal to 0 solving which you'll get lambda 0 and this is quadratic if I solve by the quadratic formula - b + - roo<unk> over b square - 4 a c by 2 a I'm going to get the answer as 0 and this is going to be 1 + minus this is 2 j and if I cancel with 2 0 1 + - j these are the three roots okay can you now tell me I will leave the same question for you as homework okay in the MCQ format please you can comment which property can help you solve this question directly without solving the characteristic equation if required I will make another video on those properties as well. Hope you got the characteristic equation format quickly.