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Expert-verified Found in: Page 309 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # There exists a ${\mathbf{4}}{\mathbf{×}}{\mathbf{4}}$ matrix whose entries are all 1 or -1 , and such that ${\mathbit{d}}{\mathbit{e}}{\mathbit{t}}{\mathbit{A}}{\mathbf{=}}{\mathbf{16}}$.

Therefore, the given statement is false.

See the step by step solution

## 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 “ $m×n$ .”

## 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 $4×4$ 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. ### Want to see more solutions like these? 