Question

Let $A = \begin{bmatrix} -11 & 8 & 16\\ 8 & -23 & 8\\ 16 & 8 & -11 \end{bmatrix}$. Find an orthogonal matrix $P$ with rational entries and a diagonal matrix $D$ such that $D = P^T AP$. $P = \begin{bmatrix} -1 & -\frac{1}{2} & 1\\ -\frac{1}{2} & 1 & 0\\ 1 & 0 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 9 & 0 & 0\\ 0 & -27 & 0\\ 0 & 0 & -27 \end{bmatrix}$

          Let $A = \begin{bmatrix} -11 & 8 & 16\\ 8 & -23 & 8\\ 16 & 8 & -11 \end{bmatrix}$. Find an orthogonal matrix $P$ with rational entries and a diagonal matrix $D$ such that $D = P^T AP$.
$P = \begin{bmatrix} -1 & -\frac{1}{2} & 1\\ -\frac{1}{2} & 1 & 0\\ 1 & 0 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 9 & 0 & 0\\ 0 & -27 & 0\\ 0 & 0 & -27 \end{bmatrix}$

Let A =
< b m a t r i x >. Find an orthogonal matrix P with rational entries and a diagonal matrix D such that D = P^T AP.
P =
< b m a t r i x >, D =
< b m a t r i x >

Added by Douglas N.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Supreeta N

06/11/2023

Step 1

To find the eigenvalues, we solve the characteristic equation det(A - λI) = 0, where I is the identity matrix. det(A - λI) = det([11-λ 8 16; 8 23-λ 8; 16 8 11-λ]) Expanding along the first row, we get: (11-λ) * det([23-λ 8; 8 11-λ]) - 8 * det([8 8; 16 11-λ]) + Show more…

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i got it wrong [11 8 16 LetA= 8 23 8 Find an orthogonal matrix P with rational entries and a diagonal matrix D such that D = pT AP. 16 8 11 1 2 P = 0 ,D= 2 1 0 1