2.4 Homework
MA 2321, Fall '03, T. Olson

For each of the following problems, be sure to indicate precisely HOW you partition the matrices and how this partition is used to find your answer.

(i)

(Problem 21 on page 140)
Notice that A²=I when $A= \left( \begin{array}{rr} 1 & 0 \\ 3 & -1 \end{array} \right)$. Use partitioned matrices to show that M²=I when

$$M = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 &-1 & 0... ... 1 & 0 & -1 & 0 \\ 0 & 1 & -3 & 1 \\ \end{array} \right).$$

(ii)

Using block matrices (and no row operations), find the inverse of

$$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 3 &-1 & 0... ... & 0 & 1 & 0 \\ 0 & 0 & 0 & 3 &-1 \\ \end{array} \right).$$

(iii)

Use block matrices to find the inverse of

$$\left( \begin{array}{rrrrrrrrrr} 1 & 0 & 0 & 0 & 0 & 0 & 0 &... ...& 5 & 5 & 5 & 5 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right).$$

(Hint: $\left( \begin{array}{rr} 1 & 0 \\ -a & 1 \end{array} \right)$ is the inverse of $\left( \begin{array}{rr} 1 & 0 \\ a & 1 \end{array} \right)$.)