Let A be the invertible matrix given by A = [2 -1; -2...

Let A be the invertible matrix given by A = [2 -1; -2 3] (a) Determine the eigenvalues and the corresponding normalized eigenvectors of the matrix A. (b) Find an invertible matrix P and a diagonal matrix D such that D = P^{-1}AP. (c) Hence, use the factorization A = PDP^{-1} to compute A^{-3}.

Transcript

00:05 In this question, we are given a matrix a which is equal to 2 minus 2 minus 1 and we need to find out the eigenvalue and normalized eigenvector for this.

00:17 So first of all, let us consider determinant of a minus lambda i where lambda will be our eigenvalue and i is the identity matrix.

00:27 So this will be equal to 2 minus lambda minus 1 as it is minus 2 and 3 minus lambda.

00:33 So if we find out the determinant of this matrix, then that will be equal to 2 minus lambda into 3 minus lambda and minus 2 into minus 2 that is plus 2.

00:43 So this will be minus of 2.

00:46 So this is equal to 0.

00:48 We find out determinant by equating this, we find out eigenvalue by equating this determinant equal to 0.

00:54 So here we will equate the determinant equal to 0 and we will be getting 2 into 3.

01:00 So that will be equal to 6.

01:04 Then minus 3 lambda, then minus 2 lambda and plus lambda square minus 2.

01:11 So that will be equal to 6 minus 5 lambda plus lambda square minus 2 or we can write it as 0 is equal to lambda square minus 5 lambda plus 4.

01:25 So we can write it as 0 is equal to lambda square minus 4 lambda minus lambda plus 4 because multiplication of minus 4 lambda and minus lambda is minus 5 lambda, oh sorry 4 and addition is minus 5 lambda.

01:40 So from here we will be getting lambda minus 4 into lambda minus 1 is equal to 0.

01:45 So that will imply lambda is equal to 4 and lambda is equal to 1 are two eigenvalues.

01:51 Now we need to find out, now we need to find out the eigenvector corresponding to these eigenvalues.

01:59 So eigenvector corresponding to lambda is equal to 4.

02:10 We will find out the eigenvector corresponding to lambda equal to 4.

02:13 So our matrix a minus lambda i that is a minus 4 i will be 2 minus 4 minus 2 minus 1 and 3 minus 4.

02:22 And let me call v as x y.

02:26 Alright.

02:27 So that should be equal to 0 0.

02:30 Now this will imply minus 2 into minus 1 minus 2 into minus 1 into x y should be equal to 0 0.

02:39 So from here we will be getting two equations minus 2 into x minus y is equal to 0 after matrix multiplication and this will be minus 2 x minus y is equal to 0...

Question

Let A be the invertible matrix given by A = [2 -1; -2 3] (a) Determine the eigenvalues and the corresponding normalized eigenvectors of the matrix A. (b) Find an invertible matrix P and a diagonal matrix D such that D = P^{-1}AP. (c) Hence, use the factorization A = PDP^{-1} to compute A^{-3}.

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