Suggested languages for you:

Americas

Europe

Problem 188

$\begin{array}{cccccccc} & /0 & 0 & 0 & 0 & 1 & 1 & 1 \mid \\ \text { Let } \mathrm{A}= & /0 & 2 & 6 & 2 & 0 & 0 & 4 \mid . \\ &/0 & 1 & 3 & 1 & 2 & 1 & 2 \mid\end{array}$ Reduce A to the Hermite normal form.

Expert verified

The Hermite normal form of matrix A is:
\(HNF(A) = \begin{bmatrix}
0 & 0 & 0 & 0 & 4 & 2 & 0 \\
0 & 1 & 3 & 1 & -2 & -1 & 2
\end{bmatrix}\)

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 8

If the method of Gauss elimination corresponds in its final form to an echelon matrix, what is the matrix analogue of the Gauss-Jordan method for solving linear systems of equations? Explain by example.

Chapter 8

[A] Show that each of the following systems has a non-zero solution: (a) \(x+2 y-3 z+w=0\) \(x-3 y+z-2 w=0\) \(2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}+5 \mathrm{w}=0\) (b) \(x+y-z=0\) \(2 \mathrm{x}-3 \mathrm{y}+\mathrm{z}=0\) \(\mathrm{x}-4 \mathrm{y}+2 \mathrm{z}=0\) [B] Show that following system has a unique solution: \(\mathrm{x}+\mathrm{y}-\mathrm{z}=0\) \(2 \mathrm{x}+4 \mathrm{y}-\mathrm{z}=0\) \(3 x+2 y+2 z=0\)

Chapter 8

Define elementary row operations and give an example,

Chapter 8

Consider the following nonhomogeneous system of linear equations. $2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}=1$ \(3 x+2 y-2 z=2\) \(\mathrm{x}+\mathrm{y}+\mathrm{z} \quad=1\) Show that (i) any two solutions to the system (1) differ by a vector which is a solution to the homogeneous system \(2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}=0\) \(3 x+2 y-2 z=0\) \(\mathrm{x}+\mathrm{y}+\mathrm{z} \quad=0\) (ii) the sum of a solution to (1) and a solution to (2) gives a solution to (1).

Chapter 8

Let \(\mathrm{V}\) be the real vector space of 2 by 2 symmetric matrices with entries in \(\mathrm{R}\). Show that \(\operatorname{dim} \mathrm{V}=3\).

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner