18.06 Problem Set #2

Due Wednesday, Feb 20

1) (5 points) Suppose A is 3 by 5, B is 5 by 3, C is 5 by 1, and D is 3 by 1. Which of these matrix operations are allowed, and what are the shape of the results?

BA, A(B+C), ABD, AC+BD, ABABD

2) (5 points) Find the inverse (assuming they exist) of these block matrices:

[ \begin{matrix}I & 0 \\ C & −I \end{matrix} ] , [ \begin{matrix}A & B \\ 0 & D \end{matrix} ] , [ \begin{matrix}0 & I \\ −I & D \end{matrix} ] .

3) (5 points) Compute L and U for the symmetric matrix

A = [ \begin{matrix}a & a & a & a \\ a & b & b & b \\ a & b & c & c \\ a & b & c & d \end{matrix} ] .

Find four conditions on a, b, c, d to get A=LU with four pivots.

4) (5 points) Which permutation matrix P makes PA upper triangular? Find permutations P_{1}, P_{2} so that P_{1}AP_{2} is lower triangular.

A = [ \begin{matrix}1 & 4 & 3 \\ 0 & 0 & 7 \\ 0 & 6 & 9 \end{matrix} ] .

5) (10 points) (a) Find missing entries:

[ \begin{matrix}1 & 3 & 0 \\ 2 & 5 & 1 \\ −3 & −9 & −1 \end{matrix} ]^{−1} = [ \begin{matrix}4 & 3 & 3 \\ ∗ & −1 & −1 \\ ∗ & ∗ & −1 \end{matrix} ] .

(b) Use above to compute the inverse of [ \begin{matrix}1 & 2 & −3 \\ 3 & 5 & −9 \\ 0 & 1 & −1 \end{matrix} ] .