91Ó°ÊÓ

Working a column at a time, compute the products $$ \left[\begin{array}{ll} 4 & 1 \\ 5 & 1 \\ 6 & 1 \end{array}\right]\left[\begin{array}{l} 1 \\ 3 \end{array}\right] \text { and }\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right]\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right] \text { and }\left[\begin{array}{ll} 4 & 3 \\ 6 & 6 \\ 8 & 9 \end{array}\right]\left[\begin{array}{l} \frac{1}{2} \\ \frac{1}{3} \end{array}\right] $$

Find two inner products and a matrix product: $$ \left[\begin{array}{lll} 1 & -2 & 7 \end{array}\right]\left[\begin{array}{r} 1 \\ -2 \\ 7 \end{array}\right] \text { and }\left[\begin{array}{lll} 1 & -2 & 7 \end{array}\right]\left[\begin{array}{l} 3 \\ 5 \\ 1 \end{array}\right] \text { and }\left[\begin{array}{r} 1 \\ -2 \\ 7 \end{array}\right]\left[\begin{array}{lll} 3 & 5 & 1 \end{array}\right] $$ The first gives the length of the vector (squared).

If the inverse of \(A^{2}\) is \(B\), show that the inverse of \(A\) is \(A B\). (Thus \(A\) is invertible whenever \(A^{2}\) is invertible.)

Find two points on the line of intersection of the three planes \(t=0\) and \(z=0\) and \(x+y+z+t=1\) in four-dimensional space.

For which numbers \(a\) does elimination break down (a) permanently, and (b) temporarily? $$ \begin{aligned} &a x+3 y=-3 \\ &4 x+6 y=6 \end{aligned} $$ Solve for \(x\) and \(y\) after fixing the second breakdown by a row exchange.

The column picture for the previous exercise (singular system) is $$ u\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right]+v\left[\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right]+w\left[\begin{array}{l} 1 \\ 3 \\ 2 \end{array}\right]=b $$ Show that the three columns on the left lie in the same plane by expressing the third column as a combination of the first two. What are all the solutions \((u, v, w)\) if \(b\) is the zero vector \((0,0,0)\) ?

Draw the two pictures in two planes for the equations \(x-2 y=0, x+y=6\).

Write down all six of the 3 by 3 permutation matrices, including \(P=I\). Identify their inverses, which are also permutation matrices. The inverses satisfy \(P P^{-1}=I\) and are on the same list.

(a) How many entries can be chosen independently in a symmetric matrix of order \(n ?\) (b) How many entries can be chosen independently in a skew-symmetric matrix \(\left(K^{\mathrm{T}}=-K\right)\) of order \(n ?\) The diagonal of \(K\) is zero!

If rows 1 and 2 are the same, how far can you get with elimination (allowing row exchange)? If columns 1 and 2 are the same, which pivot is missing? $$ \begin{array}{ll} 2 x-y+z=0 & 2 x+2 y+z=0 \\ 2 x-y+z=0 & 4 x+4 y+z=0 \\ 4 x+y+z=2 & 6 x+6 y+z=2 \end{array} $$