91Ó°ÊÓ

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

8. $\left[\begin{array}{ccccc}0& 0& 0& 0& 2\\ 1& 0& 0& 0& 3\\ 0& 1& 0& 0& 4\\ 0& 0& 1& 0& 5\\ 0& 0& 0& 1& 6\end{array}\right]$

Demonstrate the equation $\mathbf{\left|}\mathbf{d}\mathbf{e}\mathbf{t}\mathbf{A}\mathbf{\right|}\mathbf{=}\mathbf{||}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{||}\mathbf{||}\stackrel{\mathbf{→}}{\mathbf{v}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{||}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{||}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{n}}^{\mathbf{âŠ\yen }}\mathbf{||}$for a noninvertible$\mathit{n}\mathbf{×}\mathit{n}$matrix$\mathit{A}\mathbf{=}\left[{\stackrel{→}{v}}_1{\stackrel{→}{v}}_1.........{\stackrel{→}{v}}_n\right]$(Theorem 6.3.3).

The equation $\mathit{d}\mathit{e}\mathit{t}\left(-A\right)\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathit{A}$holds for all $\mathbf{6}\mathbf{×}\mathbf{6}$matrices.

If all the entries of a $\mathbf{7}\mathbf{×}\mathbf{7}$matrix A are 7, then $\mathit{d}\mathit{e}\mathit{t}\mathit{A}$ must be ${\mathbf{7}}^{\mathbf{7}}$.

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

9. $\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 2& 2& 2\\ 1& 1& 3& 3& 3\\ 1& 1& 1& 4& 4\\ 1& 1& 1& 1& 5\end{array}\right]$

If ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}$ and ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}$are linearly independent vectors in${\mathit{R}}^{\mathbf{2}}$ , what is the relationship between $\left[{\stackrel{→}{v}}_1{\stackrel{→}{v}}_2\right]$ and $\left[{\stackrel{→}{v}}_1{\stackrel{→}{v}}_2^{âŠ\yen }\right]$, where ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}^{\mathbf{âŠ\yen }}$ is the component of ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}$ orthogonal to${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}$ ?