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Q17E

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Found in: Page 309

### Linear Algebra With Applications

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

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# If two ${\mathbit{n}}{\mathbf{×}}{\mathbit{n}}$ matrices A and B are similar, then the equation ${\mathbit{d}}{\mathbit{e}}{\mathbit{t}}{\mathbit{A}}{\mathbf{=}}{\mathbit{d}}{\mathbit{e}}{\mathbit{t}}{\mathbit{B}}$ must hold.

Therefore, the given condition is satisfied. 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.

If A and B are similar, then there exists an invertible matrix T such that,

$A={T}^{-1}BT$

So,

$detA=de{t}^{-1}detBdetT\phantom{\rule{0ex}{0ex}}detA=\frac{1}{detT}detBdetT\phantom{\rule{0ex}{0ex}}detA=detB$

Therefore, the given condition is satisfied. So, the given statement is true.

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