MATH125 Lecture 26: 26
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Show that = 6 is an eigenvalue of. 2 2 and nd a basis for its eigenspace e6. We have to compute the null space of a 6i3. We see that there are two free variables implying the nullspace of a 6i3 is nonzero, hence 6 is an eigenvalue. We now know how to nd eigenvectors once we have the correspond- ing eigenvalues, but one questions remains: how to nd the eigenvalues of a given matrix. In this section we answer this question for matrices of size 2 2 and later on we consider a general case. Let a be a matrix of size 2 2. A scalar is an eigenvalue of a if and only if the null space of a i2 is nonzero. This happens if and only if the matrix a i2 is not invertible (reminder: for any invertible matrix b the nullspace nul(b) consists of zero vector only).