# What is the determinant of a matrix to a power?

Jul 20, 2015

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

#### Explanation:

A very important property of the determinant of a matrix, is that it is a so called multiplicative function. It maps a matrix of numbers to a number in such a way that for two matrices $A , B$,

$\det \left(A B\right) = \det \left(A\right) \det \left(B\right)$.

This means that for two matrices,

$\det \left({A}^{2}\right) = \det \left(A A\right)$

$= \det \left(A\right) \det \left(A\right) = \det {\left(A\right)}^{2}$,

and for three matrices,

$\det \left({A}^{3}\right) = \det \left({A}^{2} A\right)$

$= \det \left({A}^{2}\right) \det \left(A\right)$

$= \det {\left(A\right)}^{2} \det \left(A\right)$

$= \det {\left(A\right)}^{3}$

and so on.

Therefore in general $\det \left({A}^{n}\right) = \det {\left(A\right)}^{n}$ for any $n \in \mathbb{N}$.

Dec 20, 2017

$| {\boldsymbol{A}}^{n} | = | \boldsymbol{A} {|}^{n}$

#### Explanation:

Using the property:

$| \boldsymbol{A} \boldsymbol{B} | = | \boldsymbol{A} | \setminus | \boldsymbol{B} |$

Then we have:

$| {\boldsymbol{A}}^{n} | = | {\underbrace{\boldsymbol{A} \setminus \boldsymbol{A} \setminus \boldsymbol{A} \ldots \boldsymbol{A}}}_{\text{n terms}} |$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = | \boldsymbol{A} | \setminus | \boldsymbol{A} | \setminus | \boldsymbol{A} | \ldots . | \boldsymbol{A} |$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = | \boldsymbol{A} {|}^{n}$