Minus lambda minus 1 minus 4 lambda plus 8. This becomes- this becomes lambda plus 1. Let's see, these guys right here become an 8 and then Is this going to be? Going to be minus 1 times Minus this column minus thisĬolumn and then- or I shouldn't say column, So this product is lambda plusġ times lambda minus 2 times lambda minus 2. Just take this product plus this product plus this productĪnd then I subtract out this product times this product To this guy, but I think you get the idea. In my head to do this, is to use the rule of Sarrus. Of A if and only if the determinant of this matrix 0 plus or minus minus 1 isĠ plus 1, which is 1. Lambda minus minus 1- I'llĭo the diagonals here. Going to be lambda minus- let's just do it. Matrix minus A is going to be equal to- it's actually pretty straightforward to find. The identity matrix had 1'sĪcross here, so that's the only thing that becomes Going to be- times the 3 by 3 identity matrix is just With- lambda times the identity matrix is just We're going to use the 3īy 3 identity matrix. I think it was two videosĪgo or three videos ago. If- for some at non-zero vector, if and only if, theĭeterminant of lambda times the identity matrix minusĪ is equal to 0. So lambda is the eigenvalue ofĪ, if and only if, each of these steps are true. Is it's not invertible, or it has a determinant of 0. Is that its columns are not linearly independent. To be equal to 0 for some non-zero vector v. Me rewrite this over here, this equation just in a form I just factored the vector v outįrom the right-hand side of both of these guys, and Only if the 0 vector is equal to lambda times the identity Sides, rewrote v as the identity matrix times v. Going to write lambda times the identity matrix times v. Is equal to lambda- instead of writing lambda times v, I'm Let's just subtract Av from both sides- the 0 vector This is true if and only if-Īnd this is a bit of review, but I like to review it justīecause when you do this 10 years from now, I don't want you I could call it eigenvector v,īut I'll just call it for some non-zero vector v or Non-zero vector v is equal to lambda times that non-zero That it's a good bit more difficult just because the mathīecomes a little hairier. For a 2 by 2 matrix, so let's see if we can figure
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