U is a 2 x 2 orthogonal matrix of determinant -1. Find 5 · [0, 1]...
Let M=[[ 242, -84], [-84,268]]. Notice that 170 is an eigenvalue of M. Let U be an orthogonal matrix such that U-1 MU is diagonal, the first column of U has positive entries, and det(U)=1. Find (85)0.5 ?U.
Question 4. The spectral decomposition (or the orthogonal eigenvalue decomposi- tion) of a matrix A whose determinant is zero is given by A = (2) [11* • -*] +/- +] + (-1). tao ta + (e)- vv V2 for some v € Ry, and a real number c ER. (a) (5 points) Find the eigenvalues of A and the value of c. You must justify your answer. (b) (5 points) Find v. (c) (5 points) The matrix A can expressed...
(1 point) Find the determinant of the matrix [1 0 0 -2] M-1 0 3 0 To 3 0 Lo 1 -3 2 o det(M) =
Problem 8. a) Find the determinant det (A) for the matrix [1 -3 41 A 2 0 -1 1 b) Decide whether the matrix A has an inverse. If the inverse matrix A-1 exists, find its determinant det(A-1).
5. A 3 × 3 matrix is given by A=1020 -i 0 1 (a) Verify that A is hermitian. (b) Calculate Tr (A) and det (A), where det (A) represents the determinant of A (c) Find the eigenvalues of A. Check that their product and sum are consistent with Prob. (5b) (d) Write down the diagonalized version of A (e) Find the three orthonormal eigenvectors of A. (f) Construct the unitary matrix U that diagonalizes A, and show explicitly that...
(16). Determine the determinant of the following n x n matrix: 2 3 II 2 3 0 3 00 9 (17). If A= then A= 9 3 7 2 1 (18). Let A= 1 2 If x= is an eigenvector of A-1, then k = 1 2 (19). Let A € R3x3 and det(A - 1) = det(A + 1) = det(A - 21) = 0. Then det(A) = 1 3 3 2 (20). The rank of matrix A =...
Find an orthogonal basis for the column space of the matrix to the right. -1 5 5 1 -7 4 1 - 1 7 1 -3 -4 An orthogonal basis for the column space of the given matrix is O. (Type a vector or list of vectors. Use a comma to separate vectors as needed.) The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for 3 W. 6 -2 An...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Please complete 1 through 5 1. Find the determinant of the given matrix. 2. Find all t such that -2 t-2 3. Solve the given system by Cramer rule. 4. Solve the given system by Cramer rule. 5. Find the determinant of the given matrix by expansion across column 3. D 2 3 -2
(1 point) Compute the determinant of the matrix -1 -2 -4 -6 -7 -7 7 7 A= 0 0 0 0 -4 -5 7 det(A) (1 point) Find the determinant of the matrix 6 A- 6 -9 -7 det(A) (1 point) Find the determinant of the matrix 2 2 -2 B= 1 -1 2 3 -2 det (B)