Exercise 3

Inverses for certain types of matrices can be found in a relatively straightforward fashion.

1. The matrix $D=\begin{bmatrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -4 \\ \end{bmatrix}$ is called diagonal since the only nonzero entries are on the diagonal of the matrix.

   1. Find $D^{-1}$ by augmenting $D$ by the identity and finding its reduced row echelon form.

   2. Under what conditions is a diagonal matrix invertible?

   3. Explain why the inverse of a diagonal matrix is also diagonal and explain the relationship between the diagonal entries in $D$ and $D^{-1}$.

2. Consider the lower triangular matrix $L = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 3 & -4 & 1 \\ \end{bmatrix}$.

   1. Find $L^{-1}$ by augmenting $L$ by the identity and finding its reduced row echelon form.

   2. Explain why the inverse of a lower triangular matrix is also lower triangular.