**Diagonalization of a Matrix**For the given matrix $ A $:\[A = \begin{bmatrix} 4 & -6 \\ 1 & -1 \end{bmatrix}\]**Task:**Fill in each matrix for the diagonalization of matrix $ A $, represented as $ A = PDP^{-1} $.The equation is given as:\[A = \begin{bmatrix} 3 & 2 \\ \text{Ex: 5} & 1 \end{bmatrix} \begin{bmatrix} \phantom{} & \phantom{} \\ \phantom{} & 1 \end{bmatrix} \begin{bmatrix} 1 & \phantom{} \\ -1 & \phantom{} \end{bmatrix}\]**Explanation:**- The first bracketed matrix represents the matrix $ P $.- The second bracketed matrix requires you to find the missing entries to form the diagonal matrix $ D $.- The third bracketed matrix represents the inverse matrix $ P^{-1} $.**Objective:**Determine the missing values in the matrices such that the equation $ A = PDP^{-1} $ holds true.

Answered: Fill in each matrix for diagonalization…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

**Diagonalization of a Matrix**

For the given matrix $ A $:

\[
A = \begin{bmatrix} 4 & -6 \\ 1 & -1 \end{bmatrix}
\]

**Task:**

Fill in each matrix for the diagonalization of matrix $ A $, represented as $ A = PDP^{-1} $.

The equation is given as:

\[
A = \begin{bmatrix} 3 & 2 \\ \text{Ex: 5} & 1 \end{bmatrix}
\begin{bmatrix} \phantom{} & \phantom{} \\ \phantom{} & 1 \end{bmatrix}
\begin{bmatrix} 1 & \phantom{} \\ -1 & \phantom{} \end{bmatrix}
\]

**Explanation:**

- The first bracketed matrix represents the matrix $ P $.
- The second bracketed matrix requires you to find the missing entries to form the diagonal matrix $ D $.
- The third bracketed matrix represents the inverse matrix $ P^{-1} $.

**Objective:**

Determine the missing values in the matrices such that the equation $ A = PDP^{-1} $ holds true.

Expert Solution

Step 1: Given:

$A equals open square brackets table row 4 cell negative 6 end cell row 1 cell negative 1 end cell end table close square brackets$

We have to fill in each matrix for diagonalisable of the matrix.

$A equals P D P to the power of negative 1 end exponent equals open square brackets table row 3 2 row minus 1 end table close square brackets open square brackets table row minus minus row minus 1 end table close square brackets open square brackets table row 1 minus row cell negative 1 end cell minus end table close square brackets$