Find the eigen values of each matrix and determine a basis f

Find the eigen values of each matrix and determine a basis for each eigenspace.

Solution

Consider a matrix

The characteristic equation of above matrix is

The eigenvalues of given matrix are

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The eigenvector of the matrix corresponding to eigenvalue is a nonzero solution such that

The eigenvector corresponding to is

Thus, the basis for eigenspace is

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The eigenvector of the matrix corresponding to eigenvalue is a nonzero solution such that

The rref of above coefficient matrix is

The eigenvector corresponding to is

Thus, the basis for eigenspace is