Section 3.10 When an Eigenvalue is 0
Checkpoint 3.10.1.
Recall that if there are elementary matrices \(\mtx{E}_1\text{,}\) \(\mtx{E}_2\text{,...}\)\(\mtx{E}_k\) such that \(\mtx{E}_k\cdots\mtx{E}_2\mtx{E}_1\mtx{A} = \mtx{I}\) then \(\mtx{A}^{-1}= \mtx{E}_k\cdots\mtx{E}_2\mtx{E}_1\text{.}\) For \(\mtx{A} = \left[ \begin{array}{rrr} 3 \amp 0 \amp 1 \\ 0 \amp 2 \amp 1 \\ 1 \amp 1 \amp 1 \end{array} \right]\) find \(\mtx{A}^{-1}\) by starting with \([\mtx{A}:\mtx{I}]\) then using the same elementary row operation on both sides to get \([\mtx{I}:\mtx{A}^{-1}]\text{.}\) Check your answer by computing \(\mtx{A}^{-1}\mtx{A}\text{.}\)
Discussion Question 3.10.2.
Can eigenvalues be zero? What would it mean if a vector's eigenvalue were zero?
Checkpoint 3.10.3.
Compute the eigenvalues and their corresponding eigenspaces for \(\mtx{A} = \left[ \begin{array}{rr} 2 \amp -4 \\ -3 \amp 6 \end{array} \right]\text{.}\)
Checkpoint 3.10.4.
Compute the eigenvalues and their corresponding eigenspaces for \(\mtx{A} = \left[ \begin{array}{rrr} 1 \amp 1 \amp -1 \\ 1 \amp -1 \amp 1 \\ 1 \amp 1 \amp -1 \end{array} \right]\text{.}\)
Theorem 3.10.5.
An \(n \times n\) matrices \(\mtx{A}\) is invertible if and only if none of its eigenvalues are zero.
Challenge 3.10.6.
Given an invertible square matrix \(\mtx{A}\) with eigenvector \(\vect{v}\) and eigenvalue \(\lambda\text{,}\) show \(\mtx{A}^{-1}\) also has eigenvector \(\vect{v}\text{,}\)but with eigenvalue \(\frac{1}{\lambda}\text{.}\)