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Q27E

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Linear Algebra With Applications

Found in: Page 309

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

There exists a $4 \times 4$ matrix whose entries are all 1 or -1 , and such that $d e t A = 16$ .

Therefore, the given statement is false.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Matrix Definition

A matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n ” matrix, written “ $m \times n$ .”

Step 2: To check whether the given condition is true or false

We know that determinant of similar matrices is equal.

So, det (A) must be equal to, det (2A)which is false.

If A is a $4 \times 4$ matrix with entries -1 or 1 , then all 24 of the addends in the definition of can be either - 1 or 1 .

If we want det A = 16, then twenty of them must be 1, and the remaining four be -1, which is impossible.

Most popular questions for Math Textbooks

If all the entries of a square matrix are 1 or 0, then must be 1, 0, or -1.

For two invertible nxn matrices A and B , what is the relationship between $adj (A), adj (B), and a d j (A B)$ ?

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

4. $[\begin{matrix} 1 & - 1 & 2 & - 2 \\ - 1 & 2 & 1 & 6 \\ 2 & 1 & 14 & 10 \\ - 2 & 6 & 10 & 33 \end{matrix}]$

The equation $d e t (- A) = d e t A$ holds for all $6 \times 6$ matrices.

Does the following matrix have an LU factorization? See Exercises 2.4.90 and 2.4.93.

$A = [\begin{matrix} 7 & 4 & 2 \\ 5 & 3 & 1 \\ 3 & 1 & 4 \end{matrix}]$

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