Consider the following. 0 2 A = 3 -1 3 .- 2 0 1 (a) Compute the characteristic polynomial of A. (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (c) Compute the algebraic...

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Extracted text: Consider the following. 0 2 A = 3 -1 3 .- 2 0 1 (a) Compute the characteristic polynomial of A. (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (c) Compute the algebraic and geometric multiplicity of each eigenvalue.


Extracted text: Next, we want to find the eigenspaces corresponding to each eigenvalue. Recall that for the eigenvalue 1, the eigenspace is the null space of A - A1. Since the null space is defined as the solutions to (A – A1)x = 0, we create an augmented matrix with 0 constants and solve. First, let 1 = -1 to find the corresponding eigenspace. Simplify the matrix A - (-1)1. A - (-1)I = A + I 1 0 0 0 1 0 0 0 1 1 2 3 -1 3. = 2 1 2 0 3 0 3 2 0 Next, write the augmented matrix and use row operations to reduce. 2 0 0 2 2. 3 0 3 R2 R1→ R2 2 0 2 2. 0 2 R3 - R1 + R3