Question

A \( 2 \times 2 \) matrix \( P \) has an eigenvalue \( \lambda_1 = 2 \) with eigenvector \( x_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and another eigenvalue \( \lambda_2 = 5 \), with eigenvector \( x_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \). The matrix \( P \) is

• A. \( \begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix} \)
• B. \( \begin{pmatrix} 2 & 3 \\ 0 & 5 \end{pmatrix} \)
• C. \( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \)
• D. \( \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \)

Hint:

To construct a matrix from its eigenvalues and eigenvectors, use the formula \( P = V \Lambda V^{-1} \), where \( V \) is the matrix of eigenvectors and \( \Lambda \) is the diagonal matrix of eigenvalues.

Answer 1

The Correct Option is B

For a matrix \( P \), the eigenvalues and eigenvectors can be used to construct the matrix. The matrix \( P \) can be written in terms of the eigenvectors and eigenvalues as: \[ P = V \Lambda V^{-1} \] Where:
- \( \Lambda \) is the diagonal matrix with eigenvalues \( \lambda_1 \) and \( \lambda_2 \),
- \( V \) is the matrix whose columns are the eigenvectors.
The eigenvalues are \( \lambda_1 = 2 \) and \( \lambda_2 = 5 \), and the corresponding eigenvectors are \( x_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( x_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \). The matrix \( V \) will be \( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \), and the diagonal matrix \( \Lambda \) will be \( \begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix} \). Now, the matrix \( P \) is: \[ P = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 3 \\ 0 & 5 \end{pmatrix} \] Step 1: Conclusion
The matrix \( P \) is \( \begin{pmatrix} 2 & 3 \\ 0 & 5 \end{pmatrix} \), so the correct answer is (B).