Question

If$\mathit{d}\mathit{e}\mathit{t}\left(A^{10}\right)\mathbf{=}\mathbf{(}\mathbf{d}\mathbf{e}\mathbf{t}\mathbf{A}{\mathbf{)}}^{\mathbf{10}}$for all $\mathbf{10}\mathbf{\times }\mathbf{10}$matrices$\mathit{A}$.

Short answer

Therefore, the given equation satisfies the condition by using Cauchy-Binet formula.

$det\left(A^{10}\right)=(detA{)}^{10}$.

Step-by-step solution

Matrix Definition

A matrixis a set of numbers arranged in rowsand columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are rows and columns, the matrix is said to be a “ $m$by $n$” matrix, written “$m\times n$ .”

Given 

Given equations,

$\det \left(A^{10}\right)=(\det A{)}^{10}$

To check whether the given condition is true or false

Given equations,

$\det \left(A^{10}\right)=(\det A{)}^{10}$

By using Cauchy−Binet formula,

$A={\mathbf{L}}_{\mathbf{f}}\&\mathbf{B}={\mathbf{R}}_{\mathbf{g}},$for all $\mathbf{f},\mathbf{g}:\left[\mathbf{m}\right]\to \left[\mathbf{n}\right].$

Since the determinant of a permutation matrix equals the signature of the permutation, the identity follows from the fact that signatures are multiplicative.

Therefore, the given equation satisfies the condition by using Cauchy-Binet formula.