Question: 3. An eigenvalue for a matrix A has associated to it the algebraic multiplicity and the geometric multiplicity. Recall that the geometric multiplicity of an

3. An eigenvalue for a matrix A has associated to it the algebraic multiplicity and the geometric multiplicity. Recall that the geometric multiplicity of an eigenvalue A is the largest possible number of linearly independent eigenvectors of eigenvalue A. in this exercise we show an example of how to deal with geometric multiplicites. We will see more on this later on the course. Consider the following matrix 3 5 46 \"\"22 1101 (a) (2 points) Find its eigenvalues and their corresponding algebraic multiplicities. [Hint: You should get 1 and 2 as the only eigenvaluesl (b) (2 points) Suppose A is an eigenvalue and v is a corresponding eigenvector. Justify, using the properties of matrix and vector multiplication, that (A AI)" = 0. (c) (2 points) Verify that the algebraic multiplicity and the geometric multiplicity for the eigenvalue 2 coincide. (d) (3 points) Using r0111:r reduction [Causes elimination or otherwise, nd the dimension of the solution space of the equation (A ~ 1):) = 0. (e) (I point) Explain why this dimension coincides with the geometric multiplicity

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