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Q4E

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Linear Algebra With Applications

Found in: Page 34

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the rank of the matrices in 2 through 4.
4. $[\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}]$

The rank of the matrix $[\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}]$ is, $2$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find row reduce echelon form

The number of leading 1’s in $rref (A)$ represents the rank of a matrix A denoted by $rank (A)$ .

The given matrix is, $[\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}]$ .

The reduced row echelon form of the given matrix is:

$r r e f [\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}] = [\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{matrix}]$

Step 2: Determine the rank of a matrix.

The number of leading 1’s in the matrix $[\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{matrix}]$ are $2$ .

Hence, the rank of the given matrix is 2.

Most popular questions for Math Textbooks

Consider a positive definite quadratic form q on $R^{n}$ with symmetric matrix. We know that there exists an orthonormal eigenbasis for ${\vec{v}}_{1}, . . ., {\vec{v}}_{n}$ for A, with associated positive eigenvalues $λ_{1}, . . ., λ_{n}$ . Now consider the orthogonal Eigen basis ${\vec{w}}_{1}, \dots, {\vec{w}}_{n}$ , where ${\vec{w}}_{1} = \frac{1}{\sqrt{λ}} {\vec{v}}_{1}$ .

Show that $q (c_{1} {\vec{w}}_{1} + . . . . . . + c_{n} {\vec{w}}_{n}) = c_{n}^{2}$ .

Compute the products $A \vec{x}$ in Exercises 16 through 19 using paper and pencil (if the products are defined).

19. $[\begin{matrix} 1 & 1 & - 1 \\ - 5 & 1 & 1 \\ 1 & - 5 & 3 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

For which values of a, b, c, d, and e is the following matrix in reduced row-echelon form?

$A = [\begin{matrix} 0 & a & 2 & 1 & b \\ 0 & 0 & 0 & c & d \\ 0 & 0 & e & 0 & 0 \end{matrix}]$

Find the least square solutions of the system $A \overset{⇀}{x} = \overset{⇀}{b}$ where

$A = [\begin{matrix} 1 & 1 \\ 10^{- 10} & 0 \\ 0 & 10^{- 10} \end{matrix}] and \overset{⇀}{b} = [\begin{matrix} 1 \\ 10^{- 10} \\ 10^{- 10} \end{matrix}]$ .

Find the polynomial f(t) of degree 3 such that $f (1) = (1), f (2) = 5, f' (1) = 2,$ and $f' (2) = 9$ , where $f' (t)$ is the derivative of $f (t)$ . Graph this polynomial.

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