Math  /  Algebra

QuestionGiven a 2×22 \times 2 matrix that has the eigenvalues -3 and -5 , and the eigenvectors [63]\left[\begin{array}{l}6 \\ 3\end{array}\right] and [78]\left[\begin{array}{l}-7 \\ -8\end{array}\right] respectively, which of the following could represent PP and DD ? P=[6738]P=\left[\begin{array}{ll}6 & -7 \\ 3 & -8\end{array}\right] and D=[3005]D=\left[\begin{array}{cc}-3 & 0 \\ 0 & -5\end{array}\right] P=[6378]P=\left[\begin{array}{cc}6 & 3 \\ -7 & -8\end{array}\right] and D=[3005]D=\left[\begin{array}{cc}-3 & 0 \\ 0 & -5\end{array}\right] P=[7683]P=\left[\begin{array}{ll}-7 & 6 \\ -8 & 3\end{array}\right] and D=[3005]D=\left[\begin{array}{cc}-3 & 0 \\ 0 & -5\end{array}\right] P=[6738]P=\left[\begin{array}{ll}6 & -7 \\ 3 & -8\end{array}\right] and D=[5003]D=\left[\begin{array}{cc}-5 & 0 \\ 0 & -3\end{array}\right]

Studdy Solution

STEP 1

What is this asking? We're given the eigenvalues and eigenvectors of a matrix, and we need to find the matrices PP and DD that represent the eigenvectors and eigenvalues, respectively. Watch out! The order of the eigenvectors in PP must match the order of the eigenvalues in DD.

STEP 2

1. Construct P and D

STEP 3

Remember that PP is the matrix of **eigenvectors**, and DD is the **diagonal** matrix of **eigenvalues**.
The **eigenvectors** in PP form the columns, and the corresponding **eigenvalues** go in the same order down the **diagonal** of DD.
Everything else in DD is zero!

STEP 4

We're given the **eigenvectors** [63]\begin{bmatrix} 6 \\ 3 \end{bmatrix} and [78]\begin{bmatrix} -7 \\ -8 \end{bmatrix}.
These form the columns of PP.
Since the first vector corresponds to the eigenvalue 3-3 and the second corresponds to 5-5, we have P=[6738]P = \begin{bmatrix} 6 & -7 \\ 3 & -8 \end{bmatrix}.

STEP 5

The **eigenvalues** are 3-3 and 5-5.
These form the **diagonal** entries of DD, in the same order as their corresponding **eigenvectors** appear in PP.
So, we have D=[3005]D = \begin{bmatrix} -3 & 0 \\ 0 & -5 \end{bmatrix}.

STEP 6

The correct answer is P=[6738]P = \begin{bmatrix} 6 & -7 \\ 3 & -8 \end{bmatrix} and D=[3005]D = \begin{bmatrix} -3 & 0 \\ 0 & -5 \end{bmatrix}.

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