Question

Consider a $\mathbf{4}\mathbf{\times 4}$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 $\mathit{d}\mathit{e}\mathit{t}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{8}$, find the determinants in Exercises 11 through 16.

14.role="math" localid="1660114753050" $\mathit{d}\mathit{e}\mathit{t}\mathbf{\left[}\begin{array}{c}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\\ {\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{+}\mathbf{9}{\overrightarrow{\mathbf{v}}}_{\mathbf{4}}\\ {\overrightarrow{\mathbf{v}}}_{\mathbf{3}}\\ {\overrightarrow{\mathbf{v}}}_{\mathbf{4}}\end{array}\mathbf{\right]}$

Short answer

Therefore, the determinant of given determinant matrix is given by $4\times 4$,

$det\left[\begin{array}{c}{\overrightarrow{\mathrm{v}}}_1\\ {\overrightarrow{\mathrm{v}}}_2+9\overrightarrow{v_4}\\ {\overrightarrow{\mathrm{v}}}_3\\ {\overrightarrow{\mathrm{v}}}_4\end{array}\right]=8$.

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,

$$\begin{gathered} \det \left(\mathrm{A}\right)=8 \\ det\left[\begin{array}{c}{\overrightarrow{\mathrm{v}}}_1\\ {\overrightarrow{\mathrm{v}}}_2+9\overrightarrow{v_4}\\ {\overrightarrow{\mathrm{v}}}_3\\ {\overrightarrow{\mathrm{v}}}_4\end{array}\right] \end{gathered}$$

To find determinant

To determine,

$$\begin{gathered} \det \left[\begin{array}{c}{\overrightarrow{\mathrm{v}}}_1\\ {\overrightarrow{\mathrm{v}}}_2+9{\overrightarrow{v}}_3\\ {\overrightarrow{\mathrm{v}}}_3\\ {\overrightarrow{\mathrm{v}}}_4\end{array}\right]=\det \left[\begin{array}{c}{\overrightarrow{\mathrm{v}}}_1\\ {\overrightarrow{\mathrm{v}}}_2\\ {\overrightarrow{\mathrm{v}}}_3\\ {\overrightarrow{\mathrm{v}}}_4\end{array}\right]+\left[\begin{array}{c}0\\ 9{\overrightarrow{v}}_4\\ 0\\ 0\end{array}\right] \\ \det \left[\begin{array}{c}{\overrightarrow{\mathrm{v}}}_1\\ {\overrightarrow{\mathrm{v}}}_2+9\overrightarrow{v}4\\ {\overrightarrow{\mathrm{v}}}_3\\ {\overrightarrow{\mathrm{v}}}_4\end{array}\right]=\det \left(\mathrm{A}\right)+0 \\ \det \left[\begin{array}{c}{\overrightarrow{\mathrm{v}}}_1\\ {\overrightarrow{\mathrm{v}}}_2+9{\overrightarrow{v}}_4\\ {\overrightarrow{\mathrm{v}}}_3\\ {\overrightarrow{\mathrm{v}}}_4\end{array}\right]=8 \end{gathered}$$