Questions: Question 2 Does (P=left[beginarraycc-2 -4 3 6endarrayright]) diagonalize (A=left[beginarraycc-52 -36 72 50endarrayright]) ? True False

Transcript text: Question 2 Does $P=\left[\begin{array}{cc}-2 & -4 \\ 3 & 6\end{array}\right]$ diagonalize $A=\left[\begin{array}{cc}-52 & -36 \\ 72 & 50\end{array}\right]$ ? True False

Solution

Solution Steps

To determine if matrix $ P $ diagonalizes matrix $ A $, we need to check if $ P^{-1}AP $ is a diagonal matrix. If it is, then $ P $ diagonalizes $ A $.

Step 1: Calculate $ P^{-1} $

The inverse of matrix $ P $ is calculated as follows:

\[ P^{-1} = \begin{bmatrix} 9.0072 \times 10^{15} & 6.0048 \times 10^{15} \\ -4.5036 \times 10^{15} & -3.0024 \times 10^{15} \end{bmatrix} \]

Step 2: Compute $ P^{-1}AP $

Next, we compute the product $ P^{-1}AP $:

\[ D = P^{-1}AP = \begin{bmatrix} -8 & -16 \\ 4 & 8 \end{bmatrix} \]

Step 3: Check if $ D $ is Diagonal

To determine if $ D $ is a diagonal matrix, we observe its structure. A diagonal matrix has non-zero entries only on its main diagonal. Here, $ D $ has non-zero entries off the diagonal, specifically:

\[ D = \begin{bmatrix} -8 & -16 \\ 4 & 8 \end{bmatrix} \]

Since $ D $ is not diagonal, we conclude that $ P $ does not diagonalize $ A $.