Given that a 3 times 3 matrixs eigenvalues are 5 2 and 7 and

Given that a 3 times 3 matrix\'s eigenvalues are 5, -2, and 7, and the RREF for the eigenvalue 5 is [1 0 0 0 1 1 0 0 0], what is the geometric multiplicity with respect to the eigenvalue 5? What is the algebraic multiplicity? What is the geometric multiplicity? Given a 2 times 2 matrix that has the eigenvalues -1 and -2, and the [-6 2] and [-9 -9] respectively, which of the following could represent P and D? P = [-9 -6 -9 2] and D [-1 0 0 -2] P = [-6- 9 2 -9] and D = [-2 0 0 -1] P = [-6 -9 2 -9] and D = [-1 0 0 -2] P = [-6 2 -9 -9] and D = [-1 0 0 -2] Consider the weighted Euclidean inner product defined by = 5 u, v, + 6 u_2v_2 +2 u_3v_3. The generating matrix for this inner product is given by [5 0 0 0 6 0 0 0 2]. True False

Solution

Further, the geometric multiplicity of an eigenvalue of a matrix A is the dimension of its        corresponding eigenspace which is the nullspace of AI. Here, the nullspace for A-5I₃ being the solution space of (A-5I₃)X=0 is the set of solutions to the equations x =0 and y+z = 0 or, y= -z (where X = (x,y,z)^T)). Thus X = (0,-z,z) = z(0,-1,1)^T. This means that the eigenspace of the given 3x3 matrix corresponding to the eigenvalue 5 has dimension 1. Then the geometric multiplicity of the eigenvalue 5 is also 1.

2.As per the diagonalization theorem, A = PDP^-1 where D is the matrix with the eigenvalues on its leading diagonal and P is the matrix with the eigenvectors of A as its columns, in the same order. Hence, the first 3 options could be the [possible answers depending upon which eigenvector corresponds to which eigenvalue.

3. Let the given matrix be denoted by A. Since Au. Av = 25u₁v₁+36u₂v₂+4u₃v₃ 5u₁v₁+6u₂v₂+2u₃v₃ , hence < u,v> Au. Av. Therefore, A is not the generating matrix for the given Euclidean inner product space. The statement is False.