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Found in: Page 309

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# If all the diagonal entries of an ${\mathbit{n}}{\mathbf{×}}{\mathbit{n}}$ matrix ${\mathbit{A}}$ are odd integers and all the other entries are even integers, then ${\mathbit{A}}$ must be an invertible matrix.

Therefore, A is invertible. So, the given statement is true.

See the step by step solution

## Step 1: Matrix Definition

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

Using the definition of $detA$ , the product in the addend corresponding to the diagonal pattern is odd, while all other addends are even.

Thus, $detA$ is odd, so it must be different than 0.

Thus, $A$ is invertible. So, the given statement is true.