Question

Consider a $\mathit{n}\mathbf{\times }\mathit{n}$ matrix$\mathit{A}\mathbf{=}\left[{\overrightarrow{v}}_1{\overrightarrow{v}}_1....{\overrightarrow{v}}_n\right]$ . What is the relationship between the product $||{\overrightarrow{v}}_1||||{\overrightarrow{v}}_2||\mathbf{.}\mathbf{.}\mathbf{.}||{\overrightarrow{v}}_n||$ and $\mathbf{\left|}\mathbf{detA}\mathbf{\right|}$When is$\mathbf{\left|}\mathbf{detA}\mathbf{\right|}\mathbf{=}\mathbf{||}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{||}\mathbf{||}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{||}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{||}{\overrightarrow{\mathbf{v}}}_{\mathbf{n}}\mathbf{||}$ .

Short answer

The equality holds if and only if $||{\overrightarrow{v}}_i^{\perp }||=||{\overrightarrow{v}}_i^{}||,\forall i\in \left\{1,...,n\right\}$, which is if and only if all the vectors are mutually perpendicular.

Step-by-step solution

Matrix Definition. 

Matrix is aset of numbersarranged in rows and columns so asto form a rectangulararray.

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 “$m\times n$ .”

To find the relationship between the product ||v→1||||v→2||...||v→n||  and |detA|

We know that,

$\left|detA\right|=||\overrightarrow{v_1}||||{\overrightarrow{v}}_2^{\perp }||....||{\overrightarrow{v}}_n^{\perp }||$

Since$||{\overrightarrow{v}}_i^{\perp }||=||{\overrightarrow{v}}_i^{}||,\forall i\in \left\{1,...,n\right\}$ , then$\left|detA\right|\le ||{\overrightarrow{v}}_1||||{\overrightarrow{v}}_2||.....||{\overrightarrow{v}}_n||$ .

The equality holds if and only if$||{\overrightarrow{v}}_i^{\perp }||=||{\overrightarrow{v}}_i^{}||,\forall i\in \left\{1,...,n\right\}$ , which is if and only if all the vectors${\overrightarrow{v}}_i$ are mutually perpendicular.