91Ó°ÊÓ

Question: For two invertible n × n matrices A and B, determine which of the formulas stated in Exercise 67 through 75 are necessarily true.

${\left(A+B\right)}^{\mathbf{2}}\mathbf{=}{\mathit{A}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{A}\mathit{B}\mathbf{+}{\mathit{B}}^{\mathbf{2}}$

Question: Show that if A and B are nxn transition matrices, then AB will be a transition matrix as well. Hint: Use Exercise 67b.

Question: For two invertible n × n matrices Aand B, determine which of the formulas stated in Exercise 67 through 75 are necessarily true.

$\left(A−B\right)\left(A+B\right)\mathbf{=}{\mathit{A}}^{\mathbf{2}}\mathbf{−}{\mathit{B}}^{\mathbf{2}}$

Question: Consider the matrix A² in Example 4 of Section 2.3.

1. The third component of the first column ofA² is 1/4 . What does this entry mean in practical terms, that is, in terms of surfers following links in our mini-Web?
2. When is the i j thentry of A² equal to 0 ? Give your answer both in terms of paths of length 2 in the graph of the mini-Web and also in terms of surfers being able to get from page j to pageiby following two consecutive links.

Question: For two invertiblen × n matrices A and B, determine which of the formulas stated in Exercise 67 through 75 are necessarily true.

69.A + B is invertible, and ${\left(A+B\right)}^{\mathbf{−}\mathbf{1}}\mathbf{=}{\mathit{A}}^{\mathbf{−}\mathbf{1}}\mathbf{+}{\mathit{B}}^{\mathbf{−}\mathbf{1}}$.

If possible, compute the matrix products in Exercises 1 through 13, using paper and pencil.

6.$\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}$

Consider the linear transformation T from to given by

$$\begin{gathered} \mathbf{T}\left[{\mathbf{x}}_{\mathbf{1}} \\ {\mathbf{x}}_{\mathbf{2}}\right]\mathbf{=}{\mathbf{x}}_{\mathbf{1}}\left[\mathbf{1} \\ \mathbf{2} \\ \mathbf{3}\right]\mathbf{+}{\mathbf{x}}_{\mathbf{2}}\left[\mathbf{4} \\ \mathbf{5} \\ \mathbf{6}\right] \end{gathered}$$

Let Lbe the line in ${\mathit{R}}^{\mathbf{3}}$that consists of all scalar multiples of$\left[\begin{array}{l}2\\ 1\\ 2\end{array}\right]$. Find the orthogonal projection of the vector$\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]$onto line L.

Decide whether the matrices in Exercises 1 through 15 are invertible. If they are, find the inverse. Do the computations with paper and pencil. Show all your work. 6.

$\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 2\\ 1& 0& 1\end{array}\right]$

Question: For two invertible n × n matricesA andB , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.

70.A² is invertible, and${\left(A^2\right)}^{\mathbf{−}\mathbf{1}}\mathbf{=}{\left(A^{−1}\right)}^{\mathbf{2}}$