Question

Consider a $\mathbf{4}\mathbf{\times }\mathbf{4}$matrix A with rows${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{4}}$. If, find the determinants in Exercises 11 through16.

12. localid="1659509477853" $\mathit{d}\mathit{e}\mathit{t}\mathbf{\left[}\begin{array}{c}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\\ {\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\\ {\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\\ {\overrightarrow{\mathbf{v}}}_{\mathbf{4}}\end{array}\mathbf{\right]}$

Short answer

Therefore, the determinant of given determinant matrix is given by,

$det\left[\begin{array}{c}{\overrightarrow{v}}_1\\ {\overrightarrow{v}}_2\\ {\overrightarrow{v}}_3\\ {\overrightarrow{v}}_4\end{array}\right]=-8$

Step-by-step solution

Definition

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Given

Given determinant,

$$\begin{gathered} \det \left(\mathrm{A}\right)=8 \\ \det \left[\begin{array}{c}{\overrightarrow{\mathrm{v}}}_1\\ {\overrightarrow{\mathrm{v}}}_2\\ {\overrightarrow{\mathrm{v}}}_3\\ {\overrightarrow{\mathrm{v}}}_4\end{array}\right] \end{gathered}$$

To find determinant

To determine,

$$\begin{gathered} det\left[\begin{array}{c}{\overrightarrow{v}}_1\\ {\overrightarrow{v}}_2\\ {\overrightarrow{v}}_3\\ {\overrightarrow{v}}_4\end{array}\right]=-det\left[\begin{array}{c}{\overrightarrow{v}}_1\\ {\overrightarrow{v}}_2\\ {\overrightarrow{v}}_3\\ {\overrightarrow{v}}_4\end{array}\right] \\ det\left[\begin{array}{c}{\overrightarrow{v}}_1\\ {\overrightarrow{v}}_2\\ {\overrightarrow{v}}_3\\ {\overrightarrow{v}}_4\end{array}\right]=-det\left(A\right). \\ det\left[\begin{array}{c}{\overrightarrow{v}}_1\\ {\overrightarrow{v}}_2\\ {\overrightarrow{v}}_3\\ {\overrightarrow{v}}_4\end{array}\right]=-8 \end{gathered}$$