91Ó°ÊÓ

What is the relationship between the volume of the tetrahedron defined by the vectors $$\left[\begin{array}{c}a_{1} \\\a_{2} \\\1\end{array}\right], \quad\left[\begin{array}{c}b_{1} \\\b_{2} \\\1\end{array}\right], \quad\left[\begin{array}{c}c_{1} \\\c_{2} \\\1\end{array}\right]$$ and the area of the triangle with vertices \\[\left[\begin{array}{l}a_{1} \\\a_{2}\end{array}\right], \quad\left[\begin{array}{l}b_{1} \\\b_{2}\end{array}\right],\quad\left[\begin{array}{l}c_{1} \\\c_{2}\end{array}\right] ?\\]

Use Gaussian elimination to find the determinant of the matrices \(A\) in Exercises 1 through 10. $$\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 1 & 4 & 4 \\ 1 & -1 & 2 & -2 \\ 1 & -1 & 8 & -8 \end{array}\right]$$

Find the determinants of the matrices \(A\) and find out which of these matrices are invertible. $$\left[\begin{array}{lll} 6 & 0 & 0 \\ 5 & 4 & 0 \\ 3 & 2 & 1 \end{array}\right]$$

Use Gaussian elimination to find the determinant of the matrices \(A\) in Exercises 1 through 10. $$\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 1 & 4 & 4 \\ 1 & -1 & 2 & -2 \\ 1 & -1 & 8 & -8 \end{array}\right]$$

Find the determinants of the matrices \(A\) and find out which of these matrices are invertible. $$\left[\begin{array}{lll} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{array}\right]$$

Find the determinants of the matrices \(A\) and find out which of these matrices are invertible. $$\left[\begin{array}{lll} 1 & 2 & 3 \\ 1 & 1 & 1 \\ 3 & 2 & 1 \end{array}\right]$$

Use Gaussian elimination to find the determinant of the matrices \(A\) in Exercises 1 through 10. $$\left[\begin{array}{ccccc} 0 & 0 & 0 & 0 & 2 \\ 1 & 0 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 & 4 \\ 0 & 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right]$$

Use Gaussian elimination to find the determinant of the matrices \(A\) in Exercises 1 through 10. $$\left[\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 \\ 1 & 1 & 3 & 3 & 3 \\ 1 & 1 & 1 & 4 & 4 \\ 1 & 1 & 1 & 1 & 5 \end{array}\right]$$

If \(\vec{v}_{1}\) and \(\vec{v}_{2}\) are linearly independent vectors in \(\mathbb{R}^{2}\) what is the relationship between det \(\left[\vec{v}_{1} \quad \vec{v}_{2}\right]\) and \(\operatorname{det}\left[\vec{v}_{1} \quad \vec{v}_{2}^{\perp}\right],\) where \(\vec{v}_{2}^{\perp}\) is the component of \(\vec{v}_{2}\) orthogonal to \(\vec{v}_{1} ?\)

Find the determinants of the matrices \(A\) and find out which of these matrices are invertible. $$\left[\begin{array}{lll} 0 & 1 & 2 \\ 7 & 8 & 3 \\ 6 & 5 & 4 \end{array}\right]$$