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Q19E

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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 A is any n x n matrix, then ${\mathbit{d}}{\mathbit{e}}{\mathbit{t}}\left(A{A}^{T}\right){\mathbf{=}}{\mathbit{d}}{\mathbit{e}}{\mathbit{t}}\left({A}^{T}A\right)$.

Therefore, the given condition 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.

It applies,

$det\left(A{A}^{T}\right)=detAdet{A}^{T}\phantom{\rule{0ex}{0ex}}=detAdetA\left(\therefore detA=det{A}^{T}\right)\phantom{\rule{0ex}{0ex}}=det{A}^{T}detA\phantom{\rule{0ex}{0ex}}⇒det\left(A{A}^{T}\right)=det\left({A}^{T}A\right)$

Therefore, the given condition is true .

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