91Ó°ÊÓ

Find the parametric equation of the line in \(x-y-z\) space that goes through the given points. \((2,-3,1)\) and \((-5,2,1)\)

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} -1 & 4 \\ 5 & 0 \end{array}\right] $$

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for $$ A=\left[\begin{array}{ll} a & c \\ 0 & b \end{array}\right] $$

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} -2 & -1 \\ 3 & 2 \end{array}\right] $$

Find the parametric equation of the line in \(x-y-z\) space that goes through the given points. \((1,0,4)\) and \((3,2,0)\)

Where do a plane through \((1,-1,2)\) and perpendicular to \(\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]\) and a line through the points \((0,-3,2)\) and \((1,-2,3)\) intersect?

(a) Show that the eigenvalues of the matrix \(A=\left[\begin{array}{ll}a & 0 \\\ 0 & c\end{array}\right]\) are \(\lambda_{1}=a\), and \(\lambda_{2}=c\). (b) Show that the corresponding eigenvectors are \(\mathbf{v}_{1}=\left[\begin{array}{l}1 \\ 0\end{array}\right]\) and \(\mathbf{v}_{2}=\left[\begin{array}{l}0 \\ 1\end{array}\right]\).

Use the determinant to determine whether $$A=\left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right]$$ is invertible. If it is invertible, compute its inverse. In either case, solve \(A X=\mathbf{0}\).

Let $$ A=\left[\begin{array}{rr} -2 & -3 \\ -1 & 1 \end{array}\right] $$ Without explicitly computing the eigenvalues of \(A\), decide whether or not the real parts of both eigenvalues are negative.

Use the determinant to determine whether $$B=\left[\begin{array}{rr} 1 & 1 \\ -1 & 1 \end{array}\right]$$ is invertible. If it is invertible, compute its inverse. In either case. solve \(B X=\mathbf{0}\).