Questions: Quiz 8 Remaining Time: 02:19:39 Let A=[[-7, 2, 12], [-3, 0, 6], [-3, 1, 5]]. Given that λ=-1 is an eigenvalue of A, find a basis for the eigenspace of λ. Enter your basis vectors as a list of column vectors separated by commas, for example: [1, 2, 3],[4, 5, 6], etc.

Solution Steps

To find a basis for the eigenspace corresponding to the eigenvalue \(\lambda = -1\), we need to solve the equation \((A - \lambda I)\mathbf{v} = \mathbf{0}\), where \(I\) is the identity matrix of the same size as \(A\). This involves computing the matrix \(A + I\), and then finding the null space of this matrix, which will give us the basis vectors for the eigenspace.

We are given the matrix

\[ A = \begin{bmatrix} -7 & 2 & 12 \\ -3 & 0 & 6 \\ -3 & 1 & 5 \end{bmatrix} \]

and the eigenvalue \(\lambda = -1\).

To find the eigenspace corresponding to \(\lambda\), we compute the matrix \(A + I\), where \(I\) is the identity matrix:

\[ A + I = \begin{bmatrix} -7 + 1 & 2 & 12 \\ -3 & 0 + 1 & 6 \\ -3 & 1 & 5 + 1 \end{bmatrix} = \begin{bmatrix} -6 & 2 & 12 \\ -3 & 1 & 6 \\ -3 & 1 & 6 \end{bmatrix} \]

Next, we find the null space of the matrix \(A + I\). The null space consists of all vectors \(\mathbf{v}\) such that

\[ (A + I)\mathbf{v} = \mathbf{0} \]

The basis for the null space is given by the vectors:

\[ \begin{bmatrix} \frac{1}{3} \\ 1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix} \]

Final Answer