Problem
( 1 point) Find a $2 \times 2$ matrix $A$ such that \[ \left[\begin{array}{c} -2 \\ 0 \end{array}\right], \quad \text { and } \quad\left[\begin{array}{l} 1 \\ 3 \end{array}\right] \] are eigenvectors of $A$ with eigenvalues 5 and -2 , respectively.
Solution
Step 1 ：Given that the eigenvectors are \(v_1 = [-2, 0]^T\) and \(v_2 = [1, 3]^T\), and the corresponding eigenvalues are \(\lambda_1 = 5\) and \(\lambda_2 = -2\).
Step 2 ：We can form two equations from the eigenvector-eigenvalue pairs: \(A[-2, 0]^T = 5[-2, 0]^T\) and \(A[1, 3]^T = -2[1, 3]^T\).
Step 3 ：Solving these equations, we find the matrix \(A\) to be \(A = [[-10, -2], [0, -6]]\).
Step 4 ：\(\boxed{A = \left[\begin{array}{cc}-10 & -2 \\ 0 & -6\end{array}\right]}\)
