Question

There exist real invertible $\mathbf{3}\mathbf{\times }\mathbf{3}$ matrices A andSsuch that ${\mathbf{S}}^{\mathbf{T}}\mathit{A}\mathbf{S}\mathbf{=}\mathbf{-}\mathit{A}$

Short answer

Therefore, the given statement is not satisfied.

Step-by-step solution

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 “$m\times n$.”

To check whether the given condition is true or false

On taking determinant

$$\begin{gathered} det\left(S\right)det\left(A\right)det\left(S^T\right)={\left(-1\right)}^{3}det\left(A\right) \\ -{\left(det\left(S\right)\right)}^{2}detA=det\left(A\right) \end{gathered}$$,

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

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

$detA=det\left(-A\right)$

$detA=-detA$,

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