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Q32 E

Expert-verified
Found in: Page 309

### Linear Algebra With Applications

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

# There exist real invertible ${\mathbf{3}}{\mathbf{×}}{\mathbf{3}}$ matrices A and S such that .

Therefore, the given statement is not satisfied.

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

On taking determinant

$det\left(S\right)det\left(A\right)det\left({S}^{T}\right)={\left(-1\right)}^{3}det\left(A\right)\phantom{\rule{0ex}{0ex}}-{\left(det\left(S\right)\right)}^{2}detA=det\left(A\right)$,

Which is false, as det(S) turns out to be imaginary.

The question is whether there exists an invertible $3×3$ matrix A such that A and 2A are similar. If they were similar, it would have to apply

det A = det(-A)

det A = - detA,

Which, since $detA\ne 0$, is a contradiction. Therefore, the given statement is not satisfied.