91影视

in exercises 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions.

6.$$\begin{gathered} \left|x+2y+3z=8 \\ x+3y+3z=10 \\ x+2y+4z=9\right| \end{gathered}$$

An n脳nmatrix Ais said to be a Hankel matrix(named after the German mathematician Hermann Hankel, 1839鈥�1873) if ${\mathit{a}}_{\mathbf{i}\mathbf{j}}\mathbf{=}{\mathit{a}}_{\mathbf{i}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{j}\mathbf{-}\mathbf{1}}$for all $\mathit{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{n}\mathbf{-}\mathbf{1}$and all$\mathit{j}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{n}$ meaning that Ahas constant positive sloping diagonals. For example, a 4脳4 Hankel matrix is of the form.

$\mathit{A}\mathbf{=}\left[\begin{array}{cccc}a& b& c& d\\ b& c& d& e\\ c& d& e& f\\ d& e& f& g\end{array}\right]$

Show that the n脳nHankel matrices form a subspace of ${\mathit{R}}^{\mathbf{n}\mathbf{脳}\mathbf{n}}$Find the dimension of this space.

Make me a crown weighing 60 minae from a mixture of gold, bronze, tin, and wrought iron. Let the gold and bronze together form two-thirds of the weight, the gold and tin together three-fourths, and the gold and iron three-fifths. Tell me how much gold, tin, bronze, and iron you must use. (From the Greek Anthology by Metrodorus, ${\mathbf{6}}^{\mathbf{th}}$century A.D.)

Three merchants find a purse lying in the road. One merchant says, 鈥淚f I keep the purse, I will have twice as much money as the two of you together.鈥� 鈥淕ive me the purse and I will have three times as much as the two of you together,鈥� said the second merchant. The third merchant said, 鈥淚 will be much better off than either of you if I keep the purse, I will have five times as much as the two of you together.鈥� If there are coins (of equal value) in the purse, how much money does each merchant have? (From Mahavira)

In exercises 1 through 10, find all solutions of the linear systems using elimination. Then check your solutions

7. $$\begin{gathered} \left|x+2y+3z=1 \\ x+3y+4z=3 \\ x+4y+5z=4\right| \end{gathered}$$

Let U 鈮� 0 be a real upper triangular n 脳 n matrix with zeros on the diagonal. Show that (In + U) t 鈮� t n(In + U + U2 +路路路+ Un鈭�1) for all positive integers t. See Exercises 46 and 47.

Consider the matrix

$\mathbf{E}\mathbf{=}\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ 0& 0& 1\end{array}\right]$

and an arbitrary$\mathbf{3}\mathbf{脳}\mathbf{3}$matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right]$

Compute EA. Comment on the relationship between A and E A, in terms of the technique of elimination we learned in Section 1.2.

b. Consider the matrix

$\mathbf{E}\mathbf{=}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1/4& 0\\ 0& 0& 1\end{array}\right]$

and an arbitrary $\mathbf{3}\mathbf{脳}\mathbf{3}$matrix A. Compute E A. Comment on the relationship between A and E A.

c. Can you think of a $\mathbf{3}\mathbf{脳}\mathbf{3}$matrix Esuch that E A is obtained from A by swapping the last two rows (for $\mathbf{3}\mathbf{脳}\mathbf{3}$matrix A)?

d. The matrices of the forms introduced in parts (a), (b), and (c) are called elementary: An$\mathit{n}\mathbf{脳}\mathit{n}$matrix Eis elementary if it can be obtained from${\mathit{l}}_{\mathbf{n}}$by performing one of the three elementary row operations on${\mathit{l}}_{\mathbf{n}}$. Describe the format of the three types of elementary matrices.

Ifis a symmetric matrix, what can you say about the definiteness of${\mathbf{A}}^{\mathbf{2}}$? When is${\mathbf{A}}^{\mathbf{2}}$ positive definite?

In exercises, 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions.

8.$$\begin{gathered} \left|x+2y+3z=0 \\ 4x+5y+6z=0 \\ 7x+8y+10z=0\right| \end{gathered}$$

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss鈥揓ordan elimination. Show all your work.

$\left|\begin{array}{c}{x}_2+2x_4+3x_5=0\\ 4x_4+8x_5=0\end{array}\right|$