Diagonalize the following matrix the real eigenvalues are...

Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. $\begin{bmatrix} 4 & 3 & 3 \\ 3 & 4 & 3 \\ 3 & 3 & 4 \end{bmatrix}$; $\lambda = 1, 10$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. For $P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 10 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 10 \end{bmatrix}$ B. For $P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ C. The matrix cannot be diagonalized.

Transcript

00:01 Hi, now we are going to diagonalize the matrix 23, 80, minus 20, minus 10, minus 37, 10, minus 15, minus 60, 18.

00:18 So now i take this matrix as a.

00:21 So we have to find the eigenvalues and eigenvector.

00:25 For finding the eigenvalues, we have to put determinant of a minus lambda i is equal to 0.

00:35 So we get determinant of 23 minus lambda, 80 minus 20, minus 10, minus 37 minus lambda, 10, minus 15, minus 60, 18 minus lambda is equal to 0.

00:53 So from this we get the equation minus lambda cube plus 4 lambda square plus 3 lambda minus 18 is equal to 0.

01:03 On further factorization, we get lambda is equal to minus 2 and lambda is equal to 3 with multiplicity of 2.

01:21 So now we can write the matrix d is equal to lambda 1, 0, 0, 0, lambda 2, 0, 0, 0, lambda 3.

01:35 So now we have to substitute the corresponding values in this.

01:38 Then the matrix d will be equal to minus 2, 0, 0, 0, 3, 0, 0, 0, 3.

01:48 And next we have to find the eigenvector.

01:51 For finding the eigenvectors, we have to put ax is equal to lambda x.

01:59 So now i take lambda is equal to minus 2.

02:03 Then we get 25 x1 plus 80 x2 minus 20 x3 is equal to 0, minus 10 x1 minus 35 x2 plus 10 x3 is equal to 0, minus 15 x1 minus 60 x2 plus 20 x3 is equal to 0.

02:29 Now i label this one as equation number 1, this one equation number 2 and this one as equation number 3.

02:35 Now i am going to do equation 1 plus equation 3...

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