00:01
So we're giving the following matrix a.
00:03
We want to compute its character, say, polynomial.
00:06
So first, we'll compute the matrix a minus lambda i, which is 1 minus lambda, 3, negative 2, and 6 minus lambda.
00:22
So now we have 1 minus lambda times 6 minus lambda plus 6 as it's determinant.
00:32
We can expand this out to get 12 minus 7 lambda plus lambda squared as its characteristic polynomial.
00:46
And we can factor this further to get lambda minus 4 times limit minus 3.
00:54
Hence the eigenvalues are 3 and 4.
00:59
We want to compute a basis for each eigenspace.
01:02
So first we'll take the kernel of a minus 3i, which is one of our eigenvalues, and we have the matrix negative 2, 3, negative 2, 3, augmented with the 0 ,0.
01:24
We can do some basic rule operations to get negative 2, 3, 0, 0 ,000, hence we have, equals x2 times 3 for 2, 1...