Problem 4. (15 pts) Use the determinant to find out for which value(s) of the constant $k$ the $\begin{bmatrix} 1 & 2 & 3 \\ 4 & k & 5 \\ 6 & 7 & 8 \end{bmatrix}$ given matrix is invertible. Problem 5. (10 pts) (a) Show that the eigenvalues $\lambda$ of a 2x2 matrix A are the solutions of the equation $\lambda^2 - \text{tr}(A) \lambda + (\text{det } A) = 0$ (b) Assume that A is an nxn orthogonal matrix, show that the possible real eigenvalues of A are 1 and -1. (hint: A is orthogonal if (and only if) $||Ax|| = ||x||$ for all x in $R^n$)
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To do this, we can calculate the determinant of the matrix and set it equal to zero. The given matrix is: [1 2 3] [4 k 5] [L 6 7] [8] To calculate the determinant, we can use the cofactor expansion along the first row: det = 1 * (k * 7 - 5 * 6) - 2 * (4 * 7 - 5 Show more…
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Question 3 [10 points] For the matrix A below, find a value of k so that A has two basic eigenvectors associated with the eigenvalue λ = 1. A = [1 -3 k -16; 0 2 -9 10; 0 0 -1 2; 0 0 0 1] k =
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Question 7 a) Find the eigenvalues of the given matrix: (i) A = [1 -2; 2 -3] (ii) B = [4 0; 5 1] (iii) C = [0 1 3; -1 1 0; 1 1 2] (iv) C = [3 1 -3; 1 0 0; -2 1 2] b) Find the eigenvector corresponding to the given eigenvalue: (i) A = [1 1 2; 0 2 0; 2 0 -1], (λ = 2) (ii) B = [1 1 1; 1 1 1; 1 1 1], (λ = 0) c) Find the eigenvalues for the given matrices. Hence, find the eigenvectors for each eigenvalue. (i) C = [1 1 -1; 0 2 1; 0 0 3] (ii) D = [3 1 -3; 1 0 0; -2 1 2] (iii) E = [4 1 1; 1 4 1; 1 1 4]
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