(1 point) Suppose that a 2 x 2 matrix A has an eigenvalue 3...

(1 point) Suppose that a 2 x 2 matrix A has an eigenvalue 3 with corresponding eigenvector $egin{bmatrix} 1 -1 end{bmatrix}$ and an eigenvalue -2 with corresponding eigenvector $egin{bmatrix} -2 3 end{bmatrix}$. Find an invertible matrix $P$ and a diagonal matrix $D$ so that $A = PDP^{-1}$. Enter your answer as an equation of the form $A = PDP^{-1}$. You must enter a number in every answer blank for the answer evaluator to work properly. Note: In order to get credit for this problem all answers must be correct

Transcript

00:01 In this question, we are given that a is a 2x2 matrix, whose eigenvalues are 3 and negative 2, and the corresponding eigenvectors are 1 -1 and negative 2, 3.

00:11 And we are asked to write down a as p times d times p inverse first, and then multiply out these three matrices to get the matrix a itself.

00:23 So, this is basically the, we are diagonalizing the matrix a.

00:29 And in any diagonalization, the matrix in the middle, the matrix d, is simply the matrix containing the eigenvalues on the main diagonal and zeros everywhere else.

00:47 So, in our case, we are given the eigenvalues and d equals to 3 -0 and 0 -2.

00:55 The matrix p is a matrix whose columns are the eigenvectors of the matrix a.

01:02 And note that the order here is important.

01:06 Since we put lambda 1 as 3 as the first eigenvalue in the matrix d, we have to put the corresponding eigenvector as the first column in the matrix p.

01:17 So the first column is going to be 1 negative 1, and the second column is going to be negative 2, 3.

01:23 Well, this gives us the matrix p.

01:27 Now we need to find p inverse.

01:32 The formula is 1 over the determinant of p.

01:37 Multiplied by, now we need to interchange the entries on the main diagonal of the matrix p.

01:44 We are going to get 3 -1 on the main diagonal, and the entries on the other diagonal get multiplied by negative 1...

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